word

en-noun

word (countable and uncountable, plural words)

desc

Chinese Pidgin English: word, 𭉉 (Chinese spelling)

From Middle English word, from Old English word, from Proto-West Germanic *word, from Proto-Germanic *wurdą, from Proto-Indo-European *wr̥dʰh₁om. Doublet of verb and verve; further related to vrata.

you get more with a kind word and a gun than you do with a kind word alone

(group theory) A group element, expressed as a product of group elements.


free

en-adj

free (comparative freer or more free, superlative freest or most free)

freer (comparative)
more free (comparative)
freest (superlative)
most free (superlative)

From Middle English free, fre, freo, from Old English frēo (“free”), from Proto-West Germanic *frī, from Proto-Germanic *frijaz (“beloved, not in bondage”), from Proto-Indo-European *priHós (“pleased, loved”), from *preyH- (“to please, love”). Related to friend.
cognates, etc
Germanic cognates include West Frisian frij (“free”), Dutch vrij (“free”), Low German free (“free”), German frei (“free”), Danish, Swedish and Norwegian fri (“free”).
Other cognates include Sanskrit प्रिय (priyá-, “beloved”).
Germanic and Celtic are the only Indo-European language branches in which the PIE word with the meaning of "dear, beloved" acquired the additional meaning of "free" in the sense of "not in bondage". This was an extension of the idea of "characteristic of those who are dear and beloved", in other words friends and tribe members (in contrast to unfree inhabitants from other tribes and prisoners of war, many of which were among the slaves – compare the Latin use of liberi to mean both "free persons" and "children of a family").
The verb comes from Middle English freen, freoȝen, from Old English frēon, frēoġan (“to free; make free”), from Proto-West Germanic *frijōn, from Proto-Germanic *frijōną, from Proto-Indo-European *preyH-, and is cognate with German freien, Dutch vrijen, Czech přát, Serbo-Croatian prijati, Polish sprzyjać.

inh

there ain't no such thing as a free lunch

why buy the cow when you can get the milk for free

The fundamental group of the figure eight is free of rank 2.

en:unconstrained, abstractly
en:albegraically generic or universal

(algebra, of an algebraic structure) In any of various technical senses generic, universal.

(group theory, of a group) Having a set of generators which satisfy no non-trivial relations; equivalently, being the group of reduced words on a set of generators.


grupo

Borrowed from Italian gruppo, from Vulgar Latin *cruppō, itself from Proto-Germanic *kruppaz (“lump, round mass, body, crop”). Compare Renaissance Latin grupus.


base

en-noun

base (countable and uncountable, plural bases)

acid

antonym(s) of “chemical compound that will neutralize an alkali”

From Middle English base, bas, baas, from Old French base, from Latin basis, from Ancient Greek βάσις (básis). Doublet of basis and bass.

أَسَاس

something from which other things extend

(group theory) A sequence of elements not jointly stabilized by any nontrivial group element.


orbit

orbit (countable and uncountable, plural orbits)

Inherited from Middle English orbite, orbita, from Latin orbita (“course, track, impression, mark”).

គន្លង

kɔɔkŭənlɔɔng

(geometry, group theory) The subset of elements of a set X to which a given element can be moved by members of a specified group of transformations that act on X.


e

Close-mid front unrounded vowel
e
e (mathematical constant)

Pronunciation of IPA [eː]

Modification of capital letter E in uncial script, from Ancient Greek Ε (E, Epsilon).

a ∘ e = e ∘ a = a

(algebra, group theory) identity element.


additive

en-adj

additive (comparative more additive, superlative most additive)

more additive (comparative)
most additive (superlative)

From Late Latin additivus, from the participial stem of Latin addere (“to add”).

It is natural to look at a finite cyclic group as an additive group.

(group theory, of a group, semigroup, etc.) Whose operator is identified as addition.


kernel

→ Japanese: カーネル (kāneru)

From Middle English kernel, kirnel, kürnel, from Old English cyrnel, from Proto-West Germanic *kurnil, diminutive of Proto-Germanic *kurną (“seed, grain, corn”), equivalent to corn + -le. Cognate with Yiddish קערנדל (kerndl), Middle Dutch kernel, cornel, Middle High German kornel. Related also to Old Norse kjarni (“kernel”).

der

Meronyms: root, zero

If a function is continuous then its kernel is a closed set.

linear transformation
linear transformation

(mathematics, linear algebra, group theory, functional analysis) For a given function (especially a linear transformation between vector spaces or homomorphism between groups), the set of elements in the domain which are mapped to zero; (formally) given f : X → Y, the set {x ∈ X : f(x) = 0}.

jádro

(mathematics, algebra) set of elements mapped to zero


group

→ Gulf Arabic: قروب (grūp)

→ Japanese: グループ (gurūpu)

→ Korean: 그룹 (geurup)

→ Tongan: kulupu

From French groupe (“cluster, group”), from Italian gruppo, groppo (“a knot, heap, group, bag (of money)”), from Vulgar Latin *cruppo, Renaissance Latin grupus, from Frankish *krupp, from Proto-Germanic *kruppaz (“lump, round mass, body, crop”), from Proto-Indo-European *grewb- (“to crumple, bend, crawl”). In the sense of group theory coined (in French, as groupe) by Évariste Galois.
Cognate with German Kropf (“crop, craw, bunch”); Old English cropp, croppa (“cluster, bunch, sprout, flower, berry, ear of corn, crop”) (whence English crop); Dutch krop (“craw”), Icelandic kroppr (“hump, bunch”). Doublet of crop and croup.

Throughout this section, we shall assume the existence of finitely presented groups with unsolvable word problem.

1977, Roger C. Lyndon, Paul E. Schupp, Combinatorial Group Theory, Springer, page 192:

In this chapter we give some examples of Fuchsian groups. The most interesting and important ones are the so-called "arithmetic" Fuchsian groups, i.e., discrete subgroups of PSL(2,R) obtained by some "arithmetic" operations. One such construction we have already seen: if we choose all matrices of SL(2,R) with integer coefficients, then the corresponding elements of PSL(2,R) form the modular group PSL(2,Z).

1992, Svetlana Katok, Fuchsian Groups, University of Chicago Press, page 112:

2007, Zhong-Qi Ma, Group Theory for Physicists, World Scientific, page 277,
In Chap. 4 the fundamental concepts on Lie groups have been introduced through the SO(3) group and its covering group SU(2).

(group theory) A set with an associative binary operation, under which there exists an identity element, and such that each element has an inverse.


群

CJKV/code

Sino-Xenic (群):
* → Japanese: 群(ぐん) (gun)

/t͡ɕʰyə̯ɻ³⁵/

/t͡ɕʰyə̯ɻ³⁵/

Phono-semantic compound (形聲／形声, OC *ɡlun): phonetic 君 (OC *klun) + semantic 羊.
From Proto-Sino-Tibetan *m-kul (“twenty; all”); cognate with 昆 (OC *kuːn, “swarming”), Tibetan ཀུན (kun, “all”), Burmese အကုန် (a.kun, “all”) (Benedict, 1972; STEDT).


order

en-noun

order (countable and uncountable, plural orders)

→ Swahili: oda

From Middle English ordre, from Old French ordre, ordne, ordene (“order, rank”), from Latin ōrdinem, accusative of ōrdō (“row, rank, regular arrangement”, literally “row of threads in a loom”), from Proto-Italic *ordō (“to arrange”), probably ultimately from Proto-Indo-European *h₂or-d-, from *h₂er-.
Related to Latin ōrdior (“begin”, literally “begin to weave”). In sense “request for purchase”, compare bespoke. Doublet of ordo.

cog

The object of this note is to observe that it is possible to calculate the order of an element A of G#x3D;#x5C;mathit#x7B;GL#x7D;(d,q) on average using O(d³#x5C;mathsf#x7B;log#x7D;#x5C;q) field operations, assuming that qⁱ-1 has been factorised for i#x5C;led.

1997, Frank Celler, C. R. Leedham-Green, “Calculating the Order of an Invertible Matrix”, in Larry Finkelstein, William M. Kantor, editors, Groups and Computation II, American Mathematical Society, page 55:

1999, A. Ehrenfeucht, T. Harju, G. Rozenberg, The Theory of 2-structures, World Scientific, page 15,
If Δ is a finite group, its cardinality is called the order of Δ. The order of an element a∈Δ is defined as the smallest nonnegative integer n such that aⁿ=1_Δ. The second case of the following result is known as Cauchy's theorem.
Theorem 1.10 Let Δ be a finite group.
(i) The order of an element a∈Δ divides the order |Δ| of the group.
(ii) If a prime number p divides |Δ|, then there exists an element a∈Δ of order p.

2010, A. R. Vasishta, A. K. Vasishta, Modern Algebra, Krishna Prakashan Media, 60th Edition, page 180,
Since in a finite group the order of an element must be a divisor of the order of the group, therefore o (a) cannot be 3 and so we must have o (a)=4=the order of the group G.

(group theory, of an element g of a group G) The smallest positive natural number n such that (denoting the group operation multiplicatively) gⁿ is the identity element of G, if such an n exists; if no such n exists the element is said to be of infinite order (or sometimes zero order).


center

From Middle English center, centre, from Middle French centre, from Latin centrum, from Ancient Greek κέντρον (kéntron), from κεντεῖν (kenteîn, “to prick, goad”). Doublet of centrum.
* (group theory): In the notation Z(G), the letter Z derives from the German Zentrum (“center”).

sentrum

point equidistant from all points on the perimeter of a circle

مَرْكَز

point in the interior of figure with mean coordinates

(algebra, group theory, ring theory) The set of those elements (of a given algebraic structure) that commute with every other element, usually denoted Z(G). In the case of (semi-)groups, it is required that they commute under the (semi-)group operation; in the case of rings and (Lie) algebras, under mulitplication (i.e. the bracket for Lie algebras)

centro

set of elements of a group that commute with all elements


lattice

→ Japanese: ラティス (ratisu)

From Middle English latis, from Middle French lattis (“lathing”), from Old French lattis, from latte (“a lath”), from Frankish *lattā, *lattu (“a lath”), from Proto-Germanic *lattǭ, *lattō, *laþþō (“board; plank; ledge”), from Proto-Indo-European *(s)latn-, *(s)lat- (“beam; log”). Cognate with Old High German latta (“lath”), (German Latte), Old English lætt (“lath”), Middle Low German lāde (“plank, counter, sales counter”), German Laden (“shop”). More at lath.

(mathematics, algebra) In any of several technical senses, a substructure with properties analogous to those the set of points with integer coordinates bears in relation to the full Euclidian space these points are contained in.

(group theory) A discrete subgroup of Rⁿ which is isomorphic to Zⁿ (considered as an additive group) and which spans the real vector space Rⁿ.


local

en-adj

local (comparative more local, superlative most local)

more local (comparative)
most local (superlative)

From Middle English local, from Late Latin locālis (“belonging to a place”), possibly also via Old French local; ultimately from Latin locus (“a place”).
The ring-theoretic senses derive from Krull, who first referred to Noetherian commutative rings with a unique maximal ideal as "Stellenring" (Stellen (“place”) + ring) in 1938. The term was inspired by algebraic geometry, where local rings encode information about the behavior of curves (surfaces, etc.) at points; hence, describe "local" behavior.

der

(mathematics, not comparable, of a condition or property) Applying to or satisfied by substructures understood as "near points;" in particular:

(group theory, of a property of an infinite group) Satisfied by every finitely generated subgroup.

(mathematics, not comparable, of a condition or property) Detectable from the behavior of substructures understood to be "near points;" in particular:

(group theory, of a property of a finite group) Detectable from the behavior of the normalizers of the nontrivial p-subgroups.


Möbius group

Eponymous for August Ferdinand Möbius (1790–1868), a German mathematician and theoretical astronomer.

projective general linear group
projective general linear group

Möbius transformations
Möbius transformations

(group theory) The projective general linear group mathbf PGL(2,ℂ): the group of Möbius transformations on the Riemann sphere.

(group theory) For a given dimension n, the group of projective transformations leaving a particular hypersphere invariant.


Abelian

Ex. 2. Show by §§ 187, 188 that there can be no transitive Abelian group of prime degree other than the cyclic group, and that there is no irreducible Abelian equation of prime degree other than the cyclic equation.

1904, Florian Cajori, An Introduction to the Modern Theory of Equations, The MacMillan Company, page 214:

All irreducible Abelian equations of the tenth order belong to the cyclical group of order ten,[…].

1914, Charles Gustave Paul Kuschke, The Abelian Equations of the Tenth Degree Irreducible in a Given Domain of Rationality, University of California Press, page 118:

In this chapter "group" means torsion free Abelian group. The "bible" of Abelian group theory is the two volume work of Fuchs [11].

2004, Friedrich Kasch, Adolf Mader, Rings, Modules, and the Total, Springer (Birkhäuser), page 101:

(mathematics, group theory, dated) Alternative letter-case form of abelian


O

From the Etruscan letter 𐌏 (o, “o”), from the Ancient Greek letter Ο (O, “omikron”), derived from the Phoenician letter 𐤏 (ʿ, “ʿayin”), from the Egyptian hieroglyph 𓁹.

(linear algebra, group theory) orthogonal group


socle

LL-Q1860 (eng)-Flame, not lame-socle.wav

From French socle, from Italian zoccolo (“wooden shoe”), from Latin socculus, diminutive of soccus (“sock”). Doublet of zoccolo.

(group theory) The subgroup generated by the minimal normal subgroups of a given group.


retract

Partly:
* from retract (verb) (see etymology 1); and
* from Late Latin retractus (“a pulling back, retreat; refuge”), from Latin retractus (“withdrawn”), the perfect passive participle of Latin retrahō (“to draw or pull back, withdraw; to bring back; to compel to turn back; to recall; to get back, recover; to hold back, restrain, withhold; to remove, take away; to bring to light again; (Late Latin) to delay”) (see etymology 1) + -tus (suffix forming action nouns from verbs). Doublet of retrait, retreat, and ritratto.

doublet

(group theory) A subgroup of a given group such that there is a surjective endomorphism from the ambient group to the subgroup which is constant on the subgroup; in this case the subgroup is a retract of the ambient group. In symbols: H in G is a retract of G if there exists a surjective homomorphism σ from G to H with σ|_H= operatorname id.


irreducible

en-adj

irreducible (comparative more irreducible, superlative most irreducible)

more irreducible (comparative)
most irreducible (superlative)

/ˌɪɹɪˈdjuːsɪbəl/

LL-Q1860 (eng)-Flame, not lame-irreducible.wav

(group theory, of a representation) Impossible to divide further into representations of lower dimension by means of any similarity transformation.


normal subgroup

en-noun

normal subgroup (plural normal subgroups)

(group theory) A subgroup H of a group G that is invariant under conjugation; that is, for all elements h of H and for all elements g in G, the element ghg⁻¹ is in H.

normaali aliryhmä

subgroup that is invariant under conjugation


subgroup

/ˈsʌbˌɡɹuːp/

LL-Q1860 (eng)-Flame, not lame-subgroup.wav

subgrup

(group theory) group within a larger group

Much of the information about a group can be gleaned from a study of its subgroups. For these reasons it is important to study the subgroup structure of the almost simple groups, and in particular their maximal subgroups.

1990, Peter B. Kleidman, Martin W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge University Press, page 1:

A subgroup H of an algebraic group G is called algebraic if H is an algebraic subvariety of G. Algebraic subgroups defined over k (as algebraic subvarieties) are called k-subgroups. An algebraic subgroup of an algebraic group is called k-closed or closed over k (resp. k-defined or defined over k) if it is k-closed (resp. k-defined) as an algebraic subvariety.

1991, Gregori A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, page 13:

This is applied in Chapter 9 to prove the first congruence subgroup theorem, which asserts that g.z = z for all z in the center of the Drinfel'd double D(H) and all g in the principal congruence subgroup.

2012, Yorck Sommerhäuser, Yongchang Zhu, Hopf Algebras and Congruence Subgroups, American Mathematical Society, page 3:

(group theory) A subset H of a group G that is itself a group and has the same binary operation as G.


projective

en-adj

projective (comparative more projective, superlative most projective)

more projective (comparative)
most projective (superlative)

Formed by the suffixation of prōiect- (the perfect passive participial stem of the Classical Latin prōiciō, whence the English verb project) with the English -ive, forming project + -ive; however, compare the post-Classical Latin prōiectīvus (“relating to purging”).

cog

(algebra) In a technical sense, general (but not necessarily so general as to be free); involving such objects:

(group theory, of a profinite group G) Such that every epimorphism from a profinite group onto G has a right inverse which is a homomorphism.


commutator

Back-formation from commutation (in the late 19th century), as is commutate (verb) too.

(group theory) A binary map in a given group G, given by [g, h] = ghg⁻¹h⁻¹, where g and h are elements of G, which yields the group's identity if and only if the group operation commutes for g and h.


solvable

Galois theory
Solution in radicals
solvable
Évariste Galois

From solve + -able. Piecewise doublet of soluble.
The mathematical senses derive from Galois theory: Galois discovered that one could determine whether a given polynomial could be solved by radicals by studying the properties of a particular group attached to a particular field extension deriving from the polynomial in question; if the group satisfies some conditions then polynomial can be solved by radicals. Any group meeting these conditions — whether or not it arises from this process — is thus called solvable, as is any field extension giving rise to such a group. The Lie-theoretic sense is by analogy, the study of Lie algebras deriving much of its terminology from group theory. In regular use by the late 19th century.

(mathematics, group theory, of a group) Having terminating derived series; see Solvable group on Wikipedia.Wikipedia


ryhmä

ryhmä (nominative singular)
ryhmät (nominative plural)
ryhmä (accusative nominative singular)
ryhmät (accusative nominative plural)
ryhmän (accusative genitive singular)
ryhmät (accusative genitive plural)
ryhmän (genitive singular)
ryhmien (genitive plural)
ryhmäin (genitive plural rare)
ryhmää (partitive singular)
ryhmiä (partitive plural)
ryhmässä (inessive singular)
ryhmissä (inessive plural)
ryhmästä (elative singular)
ryhmistä (elative plural)
ryhmään (illative singular)
ryhmiin (illative plural)
ryhmällä (adessive singular)
ryhmillä (adessive plural)
ryhmältä (ablative singular)
ryhmiltä (ablative plural)
ryhmälle (allative singular)
ryhmille (allative plural)
ryhmänä (essive singular)
ryhminä (essive plural)
ryhmäksi (singular translative)
ryhmiksi (plural translative)
ryhmättä (abessive singular)
ryhmittä (abessive plural)
ryhmin (instructive plural)
See the possessive forms below. (comitative)
ryhmäni (nominative singular)
ryhmäni (nominative plural)
ryhmäni (accusative nominative singular)
ryhmäni (accusative nominative plural)
ryhmäni (accusative genitive singular)
ryhmäni (accusative genitive plural)
ryhmäni (genitive singular)
ryhmieni (genitive plural)
ryhmäini (genitive plural rare)
ryhmääni (partitive singular)
ryhmiäni (partitive plural)
ryhmässäni (inessive singular)
ryhmissäni (inessive plural)
ryhmästäni (elative singular)
ryhmistäni (elative plural)
ryhmääni (illative singular)
ryhmiini (illative plural)
ryhmälläni (adessive singular)
ryhmilläni (adessive plural)
ryhmältäni (ablative singular)
ryhmiltäni (ablative plural)
ryhmälleni (allative singular)
ryhmilleni (allative plural)
ryhmänäni (essive singular)
ryhminäni (essive plural)
ryhmäkseni (singular translative)
ryhmikseni (plural translative)
ryhmättäni (abessive singular)
ryhmittäni (abessive plural)
ryhmineni (comitative plural)
ryhmäsi (nominative singular)
ryhmäsi (nominative plural)
ryhmäsi (accusative nominative singular)
ryhmäsi (accusative nominative plural)
ryhmäsi (accusative genitive singular)
ryhmäsi (accusative genitive plural)
ryhmäsi (genitive singular)
ryhmiesi (genitive plural)
ryhmäisi (genitive plural rare)
ryhmääsi (partitive singular)
ryhmiäsi (partitive plural)
ryhmässäsi (inessive singular)
ryhmissäsi (inessive plural)
ryhmästäsi (elative singular)
ryhmistäsi (elative plural)
ryhmääsi (illative singular)
ryhmiisi (illative plural)
ryhmälläsi (adessive singular)
ryhmilläsi (adessive plural)
ryhmältäsi (ablative singular)
ryhmiltäsi (ablative plural)
ryhmällesi (allative singular)
ryhmillesi (allative plural)
ryhmänäsi (essive singular)
ryhminäsi (essive plural)
ryhmäksesi (singular translative)
ryhmiksesi (plural translative)
ryhmättäsi (abessive singular)
ryhmittäsi (abessive plural)
ryhminesi (comitative plural)
ryhmämme (nominative singular)
ryhmämme (nominative plural)
ryhmämme (accusative nominative singular)
ryhmämme (accusative nominative plural)
ryhmämme (accusative genitive singular)
ryhmämme (accusative genitive plural)
ryhmämme (genitive singular)
ryhmiemme (genitive plural)
ryhmäimme (genitive plural rare)
ryhmäämme (partitive singular)
ryhmiämme (partitive plural)
ryhmässämme (inessive singular)
ryhmissämme (inessive plural)
ryhmästämme (elative singular)
ryhmistämme (elative plural)
ryhmäämme (illative singular)
ryhmiimme (illative plural)
ryhmällämme (adessive singular)
ryhmillämme (adessive plural)
ryhmältämme (ablative singular)
ryhmiltämme (ablative plural)
ryhmällemme (allative singular)
ryhmillemme (allative plural)
ryhmänämme (essive singular)
ryhminämme (essive plural)
ryhmäksemme (singular translative)
ryhmiksemme (plural translative)
ryhmättämme (abessive singular)
ryhmittämme (abessive plural)
ryhminemme (comitative plural)
ryhmänne (nominative singular)
ryhmänne (nominative plural)
ryhmänne (accusative nominative singular)
ryhmänne (accusative nominative plural)
ryhmänne (accusative genitive singular)
ryhmänne (accusative genitive plural)
ryhmänne (genitive singular)
ryhmienne (genitive plural)
ryhmäinne (genitive plural rare)
ryhmäänne (partitive singular)
ryhmiänne (partitive plural)
ryhmässänne (inessive singular)
ryhmissänne (inessive plural)
ryhmästänne (elative singular)
ryhmistänne (elative plural)
ryhmäänne (illative singular)
ryhmiinne (illative plural)
ryhmällänne (adessive singular)
ryhmillänne (adessive plural)
ryhmältänne (ablative singular)
ryhmiltänne (ablative plural)
ryhmällenne (allative singular)
ryhmillenne (allative plural)
ryhmänänne (essive singular)
ryhminänne (essive plural)
ryhmäksenne (singular translative)
ryhmiksenne (plural translative)
ryhmättänne (abessive singular)
ryhmittänne (abessive plural)
ryhminenne (comitative plural)
ryhmänsä (nominative singular)
ryhmänsä (nominative plural)
ryhmänsä (accusative nominative singular)
ryhmänsä (accusative nominative plural)
ryhmänsä (accusative genitive singular)
ryhmänsä (accusative genitive plural)
ryhmänsä (genitive singular)
ryhmiensä (genitive plural)
ryhmäinsä (genitive plural rare)
ryhmäänsä (partitive singular)
ryhmiään (partitive plural)
ryhmiänsä (partitive plural)
ryhmässään (inessive singular)
ryhmässänsä (inessive singular)
ryhmissään (inessive plural)
ryhmissänsä (inessive plural)
ryhmästään (elative singular)
ryhmästänsä (elative singular)
ryhmistään (elative plural)
ryhmistänsä (elative plural)
ryhmäänsä (illative singular)
ryhmiinsä (illative plural)
ryhmällään (adessive singular)
ryhmällänsä (adessive singular)
ryhmillään (adessive plural)
ryhmillänsä (adessive plural)
ryhmältään (ablative singular)
ryhmältänsä (ablative singular)
ryhmiltään (ablative plural)
ryhmiltänsä (ablative plural)
ryhmälleen (allative singular)
ryhmällensä (allative singular)
ryhmilleen (allative plural)
ryhmillensä (allative plural)
ryhmänään (essive singular)
ryhmänänsä (essive singular)
ryhminään (essive plural)
ryhminänsä (essive plural)
ryhmäkseen (singular translative)
ryhmäksensä (singular translative)
ryhmikseen (plural translative)
ryhmiksensä (plural translative)
ryhmättään (abessive singular)
ryhmättänsä (abessive singular)
ryhmittään (abessive plural)
ryhmittänsä (abessive plural)
ryhmineen (comitative plural)
ryhminensä (comitative plural)

From Proto-Finnic *rühmä (compare Estonian rühm), borrowed from Proto-Baltic [Term?] (compare Lithuanian trūkšmas).

(group theory) group (set with an associative binary operation and certain conditions)


direct product

Theorem 3.2.3. A periodic Abelian group is the direct product of its Sylow subgroups, S(p).

1976, Marshall Hall, Jr., The Theory of Groups, 2nd edition, page 40:

(group theory) Such a set of tuples formed from two or more groups, forming another group whose group operation is the component-wise application of the original group operations and of which the original groups are normal subgroups.


coset

LL-Q1860 (eng)-Flame, not lame-coset.wav

From co- + set; apparently first used 1910 by American mathematician George Abram Miller.

1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597,
Theorem 10.5. The collection consisting of an invariant subgroup H and all its distinct cosets is itself a group, called the factor group of G, usually denoted by G/H. (Remember that the left and right cosets of an invariant subgroup are identical.) Multiplication of two cosets aH and bH is defined as the set of all distinct products z = xy, with x ∈ aH and y ∈ bH; the identity element of the factor group is the subgroup H itself.

1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,
In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.

Example 3. Let G#x3D;#x5C;mathbb#x7B;Z#x7D; (the operation is #x2B;), H#x3D;2#x5C;mathbb#x7B;Z#x7D;. Then the coset 1#x2B;2#x5C;mathbb#x7B;Z#x7D; is the set of integers of the form 1#x2B;2k where k runs through all elements of #x5C;mathbb#x7B;Z#x7D;.

2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer, page 231:

(algebra, group theory) The set that results from applying a group's binary operation with a given fixed element of the group on each element of a given subgroup.

kõrvalklass

result of applying group operation with fixed element from the parent group on each element of a subgroup


grupa

grupa (nominative singular)
grupy (nominative plural)
grupy (genitive singular)
grup (genitive plural)
grupě (dative singular)
grupám (dative plural)
grupu (accusative singular)
grupy (accusative plural)
grupo (singular vocative)
grupy (plural vocative)
grupě (locative singular)
grupách (locative plural)
grupou (instrumental singular)
grupami (instrumental plural)


grupa

pl-noun

grupa f (diminutive grupka, related adjective grupowy)

grupka (diminutive)
grupowy (adjective)
grupa (nominative singular)
grupy (nominative plural)
grupy (genitive singular)
grup (genitive plural)
grupie (dative singular)
grupom (dative plural)
grupę (accusative singular)
grupy (accusative plural)
grupą (instrumental singular)
grupami (instrumental plural)
grupie (locative singular)
grupach (locative plural)
grupo (singular vocative)
grupy (plural vocative)

→ Silesian: grupa

Internationalism; possibly borrowed from German Gruppe, French groupe, or Italian gruppo, ultimately from Latin grupus. First attested in 1765.

ref

Samuel Bogumił Linde (1807–1814) “grupa”, in Słownik języka polskiego

(group theory) group (a set with an associative binary operation, under which there exists an identity element, and such that each element has an inverse)


nilpotent

LL-Q1860 (eng)-Flame, not lame-nilpotent.wav

From nil (“not any”) + potent (“having power”) with literal meaning “having zero power” - bearing Latin roots nil and potens.
Coined in 1870, along with idempotent, by American mathematician Benjamin Peirce to describe elements of associative algebras.

compound

(mathematics, algebra) In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior.

(group theory, of a group) Admitting a central series of finite length.


podgrupa

podgrupka (diminutive)
podgrupa (nominative singular)
podgrupy (nominative plural)
podgrupy (genitive singular)
podgrup (genitive plural)
podgrupie (dative singular)
podgrupom (dative plural)
podgrupę (accusative singular)
podgrupy (accusative plural)
podgrupą (instrumental singular)
podgrupami (instrumental plural)
podgrupie (locative singular)
podgrupach (locative plural)
podgrupo (singular vocative)
podgrupy (plural vocative)

(group theory) subgroup (group within a larger group)


free group

Given a set S of "free generators" of a free group, let S⁻¹ be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let (S∪S⁻¹)^* be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form xx⁻¹ or x⁻¹x where x∈S. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation ∼ such that u∼v if and only if r(u)=r(v). Then let the underlying set of the free group generated by S be the quotient set (S∪S⁻¹)^*/∼ and let its operator be concatenation followed by reduction.

A subset A of a free group F is called "separable" when there is a non-trivial free factorization F = F₁ * F₂ such that each element of A is conjugate to an element of F₁ or of F₂.

1999, John R. Stallings, “Whitehead graphs on handlebodies”, in John Cossey, Charles F. Miller, Michael Shapiro, Walter D. Neumann, editors, Geometric Group Theory Down Under: Proceedings of a Special Year in Geometric Group Theory, Walter de Gruyter, page 317:

2002, Gilbert Baumslag, B.9 Free and Relatively Free Groups, Alexander V. Mikhalev, Günter F. Pilz, The Concise Handbook of Algebra, Kluwer Academic, page 102,
The free groups in V then all take the form H/V(H), where H is a suitably chosen absolutely free group.

The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup.

2006, Anthony W. Knapp, Basic Algebra, Springer, page 303:

(group theory) A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ.

frjáls grúpa

group whose presentation consists of generators


relator

(group theory) An expression of the identity element of a group as a product of generators, used in a presentation (type of specification) of the group.


algebraically closed

(algebra, group theory, of a group) Such that any finite set of equations and inequations has a solution.


symplectic

A calque of complex, coined by Hermann Weyl in his 1939 book The Classical Groups: Their Invariants and Representations. From Ancient Greek συμπλεκτικός (sumplektikós), from συμ (sum) (variant of σύν (sún)), + πλεκτικός (plektikós) (from πλέκω (plékō)); modelled on complex (from Latin complexus (“braided together”), from com- (“together”) + plectere (“to weave, braid”)).
The symplectic group has previously been called the line complex group.

der

συμπλεκτικός

symplectic invariant

structure that is invariant under a symplectic form (regarded as a mapping)

(group theory, of a group) Whose characteristic abelian subgroups are cyclic.


centerless

(group theory, of a group) Whose center is trivial (contains only the identity element).


additive group

additiivinen ryhmä

group or algebraic object regarded as a group with operation called addition

(mathematics) Any group, or algebraic object that can be regarded as a group, whose binary operation is called addition and denoted +; specifically:

(group theory) Any (often, but not necessarily, abelian) group whose binary operation is called addition.


semisimple

LL-Q1860 (eng)-Flame, not lame-semisimple.wav

(mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.

(group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.


semidirect

It follows from 2° that σ induces the action homomorphism θ = σπ₁, which we call semidirect. We say that σ is a homomorphic transformation of (A, U).
If θ is the semidirect action homomorphism, then it follows from 1° that the θ-module is the semidirect sum ^𝔇 of (A^× U^×) and (B, V).

1971, Transactions of the Moscow Mathematical Society, page 261:

2002, Joseph J. Rotman, Advanced Modern Algebra, American Mathematical Society, Graduate Studies in Mathematics, Volume 114, 2nd Edition, page 761,
A semidirect product is so called because a direct product G of K and Q requires, in addition to K + Q = G and K ∩ Q = {1}, that both subgroups K and Q be normal (Proposition 1.86); here, only one subgroup must be normal.

Problem 37. A semidirect sum of Lie algebras is a Lie algebra.

2012, Arkadij L. Onishchik, Ernest B. Vinberg, Lie Groups and Algebraic Groups, page 37:

(mathematics, group theory) That has the nature of, or has a nature that derives from or is analogous to, a semidirect product.


sgn

sgn(σ)

(mathematics, group theory) The sign or signature of a permutation.


malnormality

(mathematics, group theory) The property of a subgroup H of a group G where, for any x in G but not in H, H and Hˣ intersect in the identity element.


extraspecial

(group theory) Being an analogue of the Heisenberg group over a finite field whose size is a prime.


kommutaattori

kommutaattori (nominative singular)
kommutaattorit (nominative plural)
kommutaattori (accusative nominative singular)
kommutaattorit (accusative nominative plural)
kommutaattorin (accusative genitive singular)
kommutaattorit (accusative genitive plural)
kommutaattorin (genitive singular)
kommutaattorien (genitive plural)
kommutaattoria (partitive singular)
kommutaattoreja (partitive plural)
kommutaattorissa (inessive singular)
kommutaattoreissa (inessive plural)
kommutaattorista (elative singular)
kommutaattoreista (elative plural)
kommutaattoriin (illative singular)
kommutaattoreihin (illative plural)
kommutaattorilla (adessive singular)
kommutaattoreilla (adessive plural)
kommutaattorilta (ablative singular)
kommutaattoreilta (ablative plural)
kommutaattorille (allative singular)
kommutaattoreille (allative plural)
kommutaattorina (essive singular)
kommutaattoreina (essive plural)
kommutaattoriksi (singular translative)
kommutaattoreiksi (plural translative)
kommutaattoritta (abessive singular)
kommutaattoreitta (abessive plural)
kommutaattorein (instructive plural)
See the possessive forms below. (comitative)
kommutaattorini (nominative singular)
kommutaattorini (nominative plural)
kommutaattorini (accusative nominative singular)
kommutaattorini (accusative nominative plural)
kommutaattorini (accusative genitive singular)
kommutaattorini (accusative genitive plural)
kommutaattorini (genitive singular)
kommutaattorieni (genitive plural)
kommutaattoriani (partitive singular)
kommutaattorejani (partitive plural)
kommutaattorissani (inessive singular)
kommutaattoreissani (inessive plural)
kommutaattoristani (elative singular)
kommutaattoreistani (elative plural)
kommutaattoriini (illative singular)
kommutaattoreihini (illative plural)
kommutaattorillani (adessive singular)
kommutaattoreillani (adessive plural)
kommutaattoriltani (ablative singular)
kommutaattoreiltani (ablative plural)
kommutaattorilleni (allative singular)
kommutaattoreilleni (allative plural)
kommutaattorinani (essive singular)
kommutaattoreinani (essive plural)
kommutaattorikseni (singular translative)
kommutaattoreikseni (plural translative)
kommutaattorittani (abessive singular)
kommutaattoreittani (abessive plural)
kommutaattoreineni (comitative plural)
kommutaattorisi (nominative singular)
kommutaattorisi (nominative plural)
kommutaattorisi (accusative nominative singular)
kommutaattorisi (accusative nominative plural)
kommutaattorisi (accusative genitive singular)
kommutaattorisi (accusative genitive plural)
kommutaattorisi (genitive singular)
kommutaattoriesi (genitive plural)
kommutaattoriasi (partitive singular)
kommutaattorejasi (partitive plural)
kommutaattorissasi (inessive singular)
kommutaattoreissasi (inessive plural)
kommutaattoristasi (elative singular)
kommutaattoreistasi (elative plural)
kommutaattoriisi (illative singular)
kommutaattoreihisi (illative plural)
kommutaattorillasi (adessive singular)
kommutaattoreillasi (adessive plural)
kommutaattoriltasi (ablative singular)
kommutaattoreiltasi (ablative plural)
kommutaattorillesi (allative singular)
kommutaattoreillesi (allative plural)
kommutaattorinasi (essive singular)
kommutaattoreinasi (essive plural)
kommutaattoriksesi (singular translative)
kommutaattoreiksesi (plural translative)
kommutaattorittasi (abessive singular)
kommutaattoreittasi (abessive plural)
kommutaattoreinesi (comitative plural)
kommutaattorimme (nominative singular)
kommutaattorimme (nominative plural)
kommutaattorimme (accusative nominative singular)
kommutaattorimme (accusative nominative plural)
kommutaattorimme (accusative genitive singular)
kommutaattorimme (accusative genitive plural)
kommutaattorimme (genitive singular)
kommutaattoriemme (genitive plural)
kommutaattoriamme (partitive singular)
kommutaattorejamme (partitive plural)
kommutaattorissamme (inessive singular)
kommutaattoreissamme (inessive plural)
kommutaattoristamme (elative singular)
kommutaattoreistamme (elative plural)
kommutaattoriimme (illative singular)
kommutaattoreihimme (illative plural)
kommutaattorillamme (adessive singular)
kommutaattoreillamme (adessive plural)
kommutaattoriltamme (ablative singular)
kommutaattoreiltamme (ablative plural)
kommutaattorillemme (allative singular)
kommutaattoreillemme (allative plural)
kommutaattorinamme (essive singular)
kommutaattoreinamme (essive plural)
kommutaattoriksemme (singular translative)
kommutaattoreiksemme (plural translative)
kommutaattorittamme (abessive singular)
kommutaattoreittamme (abessive plural)
kommutaattoreinemme (comitative plural)
kommutaattorinne (nominative singular)
kommutaattorinne (nominative plural)
kommutaattorinne (accusative nominative singular)
kommutaattorinne (accusative nominative plural)
kommutaattorinne (accusative genitive singular)
kommutaattorinne (accusative genitive plural)
kommutaattorinne (genitive singular)
kommutaattorienne (genitive plural)
kommutaattorianne (partitive singular)
kommutaattorejanne (partitive plural)
kommutaattorissanne (inessive singular)
kommutaattoreissanne (inessive plural)
kommutaattoristanne (elative singular)
kommutaattoreistanne (elative plural)
kommutaattoriinne (illative singular)
kommutaattoreihinne (illative plural)
kommutaattorillanne (adessive singular)
kommutaattoreillanne (adessive plural)
kommutaattoriltanne (ablative singular)
kommutaattoreiltanne (ablative plural)
kommutaattorillenne (allative singular)
kommutaattoreillenne (allative plural)
kommutaattorinanne (essive singular)
kommutaattoreinanne (essive plural)
kommutaattoriksenne (singular translative)
kommutaattoreiksenne (plural translative)
kommutaattorittanne (abessive singular)
kommutaattoreittanne (abessive plural)
kommutaattoreinenne (comitative plural)
kommutaattorinsa (nominative singular)
kommutaattorinsa (nominative plural)
kommutaattorinsa (accusative nominative singular)
kommutaattorinsa (accusative nominative plural)
kommutaattorinsa (accusative genitive singular)
kommutaattorinsa (accusative genitive plural)
kommutaattorinsa (genitive singular)
kommutaattoriensa (genitive plural)
kommutaattoriaan (partitive singular)
kommutaattoriansa (partitive singular)
kommutaattorejaan (partitive plural)
kommutaattorejansa (partitive plural)
kommutaattorissaan (inessive singular)
kommutaattorissansa (inessive singular)
kommutaattoreissaan (inessive plural)
kommutaattoreissansa (inessive plural)
kommutaattoristaan (elative singular)
kommutaattoristansa (elative singular)
kommutaattoreistaan (elative plural)
kommutaattoreistansa (elative plural)
kommutaattoriinsa (illative singular)
kommutaattoreihinsa (illative plural)
kommutaattorillaan (adessive singular)
kommutaattorillansa (adessive singular)
kommutaattoreillaan (adessive plural)
kommutaattoreillansa (adessive plural)
kommutaattoriltaan (ablative singular)
kommutaattoriltansa (ablative singular)
kommutaattoreiltaan (ablative plural)
kommutaattoreiltansa (ablative plural)
kommutaattorilleen (allative singular)
kommutaattorillensa (allative singular)
kommutaattoreilleen (allative plural)
kommutaattoreillensa (allative plural)
kommutaattorinaan (essive singular)
kommutaattorinansa (essive singular)
kommutaattoreinaan (essive plural)
kommutaattoreinansa (essive plural)
kommutaattorikseen (singular translative)
kommutaattoriksensa (singular translative)
kommutaattoreikseen (plural translative)
kommutaattoreiksensa (plural translative)
kommutaattorittaan (abessive singular)
kommutaattorittansa (abessive singular)
kommutaattoreittaan (abessive plural)
kommutaattoreittansa (abessive plural)
kommutaattoreineen (comitative plural)
kommutaattoreinensa (comitative plural)

[ˈko̞mːuˌt̪ɑ̝ːt̪ːo̞ri]

Internationalism (see English commutator).


grúpa

is-noun

grúpa f (genitive singular grúpu, nominative plural grúpur)

grúpu (genitive singular)
grúpur (nominative plural)
grúpa (indefinite nominative singular)
grúpan (definite nominative singular)
grúpur (indefinite nominative plural)
grúpurnar (definite nominative plural)
grúpu (accusative indefinite singular)
grúpuna (accusative definite singular)
grúpur (accusative indefinite plural)
grúpurnar (accusative definite plural)
grúpu (dative indefinite singular)
grúpunni (dative definite singular)
grúpum (dative indefinite plural)
grúpunum (dative definite plural)
grúpu (genitive indefinite singular)
grúpunnar (definite genitive singular)
grúpna (genitive indefinite plural)
grúpnanna (definite genitive plural)


grúpufræði

grúpufræði f (genitive singular grúpufræði, no plural)

grúpufræði (genitive singular)
grúpufræði (indefinite nominative singular)
grúpufræðin (definite nominative singular)
grúpufræði (accusative indefinite singular)
grúpufræðina (accusative definite singular)
grúpufræði (dative indefinite singular)
grúpufræðinni (dative definite singular)
grúpufræði (genitive indefinite singular)
grúpufræðinnar (definite genitive singular)

From grúpa (“group”) + -fræði (“-logy”).


group action

en-noun

group action (countable and uncountable, plural group actions)

(group theory) A mapping of a mathematical group to the permutations of an object that is compatible with the group operation.


hlutgrúpa

is-noun

hlutgrúpa f (genitive singular hlutgrúpu, nominative plural hlutgrúpur)

hlutgrúpu (genitive singular)
hlutgrúpur (nominative plural)
hlutgrúpa (indefinite nominative singular)
hlutgrúpan (definite nominative singular)
hlutgrúpur (indefinite nominative plural)
hlutgrúpurnar (definite nominative plural)
hlutgrúpu (accusative indefinite singular)
hlutgrúpuna (accusative definite singular)
hlutgrúpur (accusative indefinite plural)
hlutgrúpurnar (accusative definite plural)
hlutgrúpu (dative indefinite singular)
hlutgrúpunni (dative definite singular)
hlutgrúpum (dative indefinite plural)
hlutgrúpunum (dative definite plural)
hlutgrúpu (genitive indefinite singular)
hlutgrúpunnar (definite genitive singular)
hlutgrúpna (genitive indefinite plural)
hlutgrúpnanna (definite genitive plural)

From hlut- (“partial, sub”) + grúpa (“group”).

prefix

Það er alþekkt staðreynd um granngrúpur að opin hlutgrúpa er líka lokuð.

2003, Rögnvaldur G. Möller, “Umraðanagrúpur, granngrúpur og net”, in Tímarit um raunvísindi og stærðfræði, volume 1, number 2, page 7:

Sérhverja endanlega víxlna grúpu G má skrifa sem [innra] beint margfeldi G = U₁ × … × Uᵣ af rásuðum hlutgrúpum U₁,….Uᵣ.

2006, Guðmundur Einarsson, Algebra I - 11. kafli, page 1:

Losaraleg samlíking er að bera hjátrefjanir A ⊂ X saman við normlegar hlutgrúpur N ◁ G.

2008, Johann Guðmundsson, “Spor, nykrar og kyrrapunktar”, in Tímarit um raunvísindi og stærðfræði, volume 5, number 1, page 75:

normleg hlutgrúpa

distinguished subgroup, invariant subgroup, normal subgroup


trivial group

identity element
identity element#English

(group theory) The unique group (up to isomorphism) consisting of a single element (which is the identity element).


quotient group

This paper shows how to compute the quotient groups Jⁿ/Jⁿ⁺¹ (as well as the multiplicative structure of the graded ring consisting of these quotient groups).

1975, John R. Stallings, “Quotients of the Powers of the Augmentation Ideal in a Group Ring”, in Lee Paul Neuwirth, editor, Knots, Groups, and 3-manifolds: Papers Dedicated to the Memory of R. H. Fox, Princeton University Press, page 101:

The Weyl group is the quotient group N_H/T_H, and in the present case the Weyl group is simply the permutation group S₃.

1983, David H. Sattinger, Branching in the Presence of Symmetry, Society for Industrial and Applied Mathematics, page 33:

The class of reflexive groups doesn't behave nicely with regards to operations: a closed subgroup of a reflexive group need not be reflexive, and a quotient group of a reflexive group need not be reflexive.

2002, Alexander Arhangel'skii, “Topological Invariants in Algebraic Environment”, in Miroslav Hušek, Jan van Mill, editors, Recent Progress in General Topology II, North-Holland: Elsevier, page 39:

equivalence relation
equivalence relation

(group theory) A group obtained from a larger group by aggregating elements via an equivalence relation that preserves group structure.


simple group

1989, Roger W. Carter, Simple Groups of Lie Type, John Wiley & Sons, Wiley Classics Edition, page 1,
Since the first edition of this book was published in 1972, great progress has been made in the theory of finite simple groups. Above all, the classification of the finite simple groups was finally completed in 1981. […] In addition, the five Mathieu groups have been supplemented by the discovery between 1965 and 1981 of 21 further 'sporadic' simple groups.

Let #x5C;mathcal#x7B;K#x7D; be the list of finite simple groups appearing in section 47.[…]
Classification Theorem. Every finite simple group is isomorphic to a member of #x5C;mathcal#x7B;K#x7D;.

2000, Michael Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, page 260:

However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify continuous simple groups.[…]Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from #x5C;mathbb#x7B;R#x7D;, #x5C;mathbb#x7B;C#x7D;, #x5C;mathbb#x7B;H#x7D;, or #x5C;mathbb#x7B;O#x7D;.

2010, John Stillwell, Mathematics and Its History, 3rd edition, Springer, page 495:

normal subgroups
normal subgroup#English

(group theory) A group which has no normal subgroups apart from the trivial group and itself.

gruppo semplice

group having no normal subgroups other than the trivial group and itself


dihedral group

From di- and -hedron; so called because the polygon, when it can be reflected along certain axes of symmetry, has two faces: its upper side and its underside.

(group theory) a group of symmetries of a regular polygon


alternating group

en-noun

alternating group (plural alternating groups)

even permutations
even permutation#English

(group theory) a group of even permutations of a finite set


braid group

(group theory) A group whose elements represent ways to weave some number of strings into braids.


permutation group

permutation group (plural permutation groups)

In this chapter we shall be concerned with the relationship between permutation groups and graphs. We begin by explaining how a transitive permutation group may be represented graphically, and then we reverse the process, showing that a graph gives rise to a permutation group.

1979, Norman L. Biggs, A. T. White, Permutation Groups and Combinatorial Structures, page 80:

The Galois group G(L_f /C(x)) is called the monodromy group of f, denoted Mon(f), and viewed as a permutation group on the conjugates of y over C(x).

1996, Helmut Volklein, Groups as Galois Groups: An Introduction, page 47:

Now, groups are axiomatically defined, and the above concept is a permutation group, that is, a subgroup of the symmetric group.[…]The study of finite permutation groups is one of the oldest parts of group theory, motivated initially by its connection with solvability of equations.

2002, Peter J. Cameron, “B.5 Permutation Groups”, in Alexander V. Mikhalev, Günter F. Pilz, editors, The Concise Handbook of Algebra, page 86:

(algebra, group theory) A group whose elements are permutations (self-bijections) of a given set and whose group operation is function composition.

transformation group

group whose elements are permutations of a set


cyclic group

1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22,
Proposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group.

If u#x3A;G#x5F;1#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; and v#x3A;G#x5F;2#x5C;rightarrow#x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D; are isomorphisms of two cyclic groups with #x5C;mathbb#x7B;Z#x7D;#x2F;m#x5C;mathbb#x7B;Z#x7D;, then v#x7B;-1#x7D;#x5C;circu#x3A;G#x5F;1#x5C;rightarrowG#x5F;2 is an isomorphism.

2002, Serge Lang, Algebra, 3rd edition, Springer, page 24:

2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152,
Cyclic groups are the simplest groups imaginable.

(group theory) A group generated by a single element.


sporadic group

The earliest usage is thought to be that of English mathematician William Burnside in 1911, W. Burnside, Theory of Groups of Finite Order, 2nd Edition, in a comment about the Mathieu groups.

By the classification of finite simple groups in 1981 it is now known that every finite simple group is either cyclic of prime order, or an alternating group of degree 5 or bigger, or a simple group of Lie type as above, or one of 26 sporadic groups.

1998, Meinolf Geck, “Finite Groups of Lie Type”, in Roger William Carter, Meinolf Geck, editors, Representations of Reductive Groups, Cambridge University Press, page 63:

The first five sporadic groups were discovered by Mathieu in the late nineteenth century. The remaining twenty-one were discovered in the nineteen sixties and seventies.

1998, David J. Benson, “Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants”, in Peter H. Kropholler, Graham A. Niblo, Ralph Stöhr, editors, Geometry and Cohomology in Group Theory, Cambridge University Press, page 10:

It is natural to expect that the sporadic groups should play a role in the structure of the Coker(J) space, though we are only beginning to understand some of the smaller sporadic groups in this framework.

2004, Alejandro Adem, R. James Milgram, Cohomology of Finite Groups, 2nd edition, Springer, page 245:

(group theory) Any one of the 26 exceptional finite simple groups, which do not belong to any of the general, infinite categories specified by the classification theorem for finite simple groups.

Mathieu groups
William Burnside
sporadic group


seminormal

LL-Q1860 (eng)-Flame, not lame-seminormal.wav

(group theory) Of a subgroup A of a group G, having a subgroup B such that AB = G, and for any proper subgroup C of B, AC is a proper subgroup of G.


semipermutable

Every permutable subgroup of a finite group is semipermutable. The converse, however, is not necessarily true.

(group theory, of a subgroup) That commutes with every subgroup whose order is relatively prime to its own.


special unitary group

special unitary group (plural special unitary groups)

1992 [Prentice-Hall], George H. Duffey, Applied Group Theory: For Physicists and Chemists, 2015, Dover, Unabridged Republication, page 284,
The special unitary group in two dimensions is represented by the 2 X 2 unitary matrices whose determinants equal 1.

The group is called the special unitary group in two dimensions, or SU(2), because it acts on matrices of degree 2.

2000, Herbert S. Green, Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process, Springer, page 26:

2004, Roger Cooke (translator), Vladimir I. Arnold, Lectures on Partial Differential Equations, [1997, Lekstii ob uravneniyakh s chastnymi proizvodnymi], Springer, page 81, When n = 3, the group of rotations SO(3) is isomorphic to the real three-dimensional projective space ℝP³. It has a two-sheeted covering by the three-dimensional sphere (the group of unit quaternions), which in turn is isomorphic to the special unitary group SU(2), also known as the spin group of order 3, as in the following diagram

(linear algebra, group theory) For given n, the group of n×n unitary matrices with complex elements and determinant equal to one.

erityinen unitaarinen ryhmä

group of n×n matrices with complex elements and determinant 1


מסלול

he-noun

מַסְלוּל • (maslúl) m (plural indefinite מַסְלוּלִים, singular construct מַסְלוּל־, plural construct מַסְלוּלֵי־)

מַסְלוּל (canonical)
maslúl (romanization)
מַסְלוּלִים (indefinite plural)
מַסְלוּל־ (construct singular)
מַסְלוּלֵי־ (construct plural)


polynormal

(group theory, of a subgroup) Such that its closure under conjugation by any element of the parent group can also be achieved via closure by conjugation by some element in the subgroup generated.


pronormal

(group theory, of a subgroup) Such that each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate.


Coxeter group

Named for British-born Canadian mathematician H. S. M. Coxeter (1907–2003).

We may ask in general which Coxeter groups arise as subgroups of a given Coxeter group. This question is of course far too general. However, there are Coxeter groups which arise canonically as subgroups of a given Coxeter group.

1993, B. Mühlherr, “Coxeter groups in Coxeter groups”, in Albrecht Beutelspacher, F. de Clerck, editors, Finite Geometries and Combinatorics, page 277:

The above definitions, of Cayley graph, reflection, wall, half-space, folding, Bruhat order, etc., work for any Coxeter group.

2007, Michael B. Smythe, Julian Webster, “12: Discrete Spatial Models”, in Marco Aiello, Ian Pratt-Hartmann, Johan van Benthem, editors, Handbook of Spatial Logics, page 795:

The problem of classifying all reflective lattices of given rank is also of interest in its own right from the perspectives of Coxeter groups and arithmetic subgroups of O(n,1).

2012, Daniel Allcock, “The Reflective Lorentzian Lattices of Rank 3”, in Memoirs of the American Mathematical Society, number 1033, page vii:

(mathematics, geometry, group theory) Any of a class of groups whose finite cases are precisely the finite reflection groups (including the symmetry groups of polytopes), but which are more varied in their infinite cases, and whose range of application encompasses various areas of mathematics.

Coxeter group
Harold Scott MacDonald Coxeter


symmetry group

[The] situation is more delicate in the case of higher dimensional symmetry groups; it is not in general possible to reduce the order of an equation invariant under an r-parameter symmetry group by r using only quadratures.

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, published 2000, page 76:

It is often assumed that the design unit inside each fundamental region (particularly for monotranslational symmetry groups p111 and p1a1 and ditranslational symmetry groups p1 and pg) is asymmetric.

2000, C. E. Horne, Geometric Symmetry in Patterns and Tilings, page 79:

The structure of the symmetry group of a molecule affects several spectroscopic aspects and vice versa. Thus, it is clearly important to know the possible symmetries of fullerenes and similar structures.
In this chapter, we study symmetry groups of fulleroids.

2010, Stanislav Jendrol, František Kardoš, “28: Symmetry of Fulleroids”, in Klaus D. Sattler, editor, Handbook of Nanophysics: Clusters and Fullerenes, pages 28–1:

(geometry, algebra, group theory) A group whose elements are all the transformations under which a given object remains invariant and whose group operation is function composition.

full symmetry group

emphasises the inclusion of orientation-reversing transformations


semidirect product

en-noun

semidirect product (plural semidirect products)

Refers to the fact that the criteria are less strict than for the direct product; compare semidirect.

1998, Hernán Cendra, Darryl D. Holm, Jerrold E. Marsden, Tudor S. Ratiu, Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products, A. G. Khovanskiĭ, A. Varchenko, V. Vassiliev (editors), Geometry of Differential Equations, American Mathematical Society, Translations, Series 2, Volume 186, Advances in the Mathematical Sciences 39, page 8,
The preceding result is a special case of a general theorem on reduction by stages for semidirect products acting on a symplectic manifold […] .

Also, by Lemma 3.1, every group G with a normal subgroup N and complement H is isomorphic to a semidirect product of N by H, and so once we prove Theorem 3.2, it will be fair to say that we have constructed all possible split extensions (up to isomorphism).

2008, I. Martin Isaacs, Finite Group Theory, American Mathematical Society, page 69:

Theorem 4.8.5 Let G be a group, H,K#x3C;G,H◅ G. Then G is the internal semidirect product of H and K if and only if H#x5C;capK#x3D;1 and G#x3D;HK.
Many groups can be understood as semidirect products.

2012, Fernando Q. Gouvêa, A Guide to Groups, Rings, and Fields, page 61:

(group theory) A generalisation of direct product such that, in one of two equivalent definitions, only one of the subgroups involved is required to be a normal subgroup.


inverse semigroup

inverse semigroup (plural inverse semigroups)

Examples of regular semigroups include any band, inverse semigroup or completely regular semigroup (see sections A. 2, A. 11, and A. 10); in particular, any group is a regular semigroup.

2002, Peter G. Trotter, “A.9: Regular Semigroups”, in Aleksandr Vasilʹevich Mikhalev, Günter Pilz, editors, The Concise Handbook of Algebra, page 35:

Inverse semigroups form a class of semigroups which are closest (in some sense) to groups.

2008, Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, page v:

Inverse semigroups are central to modern semigroup theory: arguably, they form the most-studied class of semigroups.

2014, Christopher Hollings, Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, page 249:

(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.

inverse Halbgruppe

semigroup in which every element has an inverse


Futurama theorem

From "The Prisoner of Benda" (2010), season 6, episode 10 of Futurama. The episode and the theorem were written by the mathematician Ken Keeler.

(group theory) A theorem which proves that, given a mind swap machine that can't be used on the same pair of people more than once, it is possible to reverse any number of mind swaps by introducing two more people that did not have their minds swapped with anyone before.

Futurama
Ken Keeler
The Prisoner of Benda


(group theory, uncommon) Denotes the graphical equality of two strings of symbols.


子群

/t͡sz̩²¹⁴⁻²¹ t͡ɕʰyn³⁵/

/t͡siː³⁵ kʷʰɐn²¹/

/t͡sz̩²¹⁴⁻²¹ t͡ɕʰyn³⁵/

/t͡siː³⁵ kʷʰɐn²¹/


Cayley's theorem

(group theory) The theorem stating that every group G is isomorphic to a subgroup of the symmetric group acting on G.


special linear group

special linear group (plural special linear groups)

The complex special linear group SL(n, C) is the subgroup of GL(n, C) consisting of matrices with determinant one.[…]The special linear groups are sometimes also called unimodular groups.

1972, Johan G. F. Belinfante, Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods, Society for Industrial and Applied Mathematics, page 6:

1998, F. Celler, C. R. Leedham-Green, A constructive recognition algorithm for the special linear group, Robert Curtis, Robert Wilson (editors), The Atlas of Finite Groups: Ten Years On, Cambridge University Press, 2003 Digitally Printed Edition, page 11,
In the first part of this note we present an algorithm to recognise constructively the special linear group.

Hence, all congruent bases induced from a game context belong to the special linear group SL(m).

2014, Holger Ingmar Meinhardt, The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, Springer, page 100:

(group theory) For given field F and order n, the group of n×n matrices with determinant 1, with the group operations of matrix multiplication and matrix inversion.


general linear group

general linear group (plural general linear groups)

1993, Peter J. Olver, Applications of Lie Groups to Differential Equations, Springer, 2000, Softcover Reprint, page 17,
Often Lie groups arise as subgroups of certain larger Lie groups; for example, the orthogonal groups are subgroups of the general linear groups of all invertible matrices.

We will again call this group the general linear group and denote it by GL₃.
In just the same way, the set of all nonsingular linear transformations of the plane L₂ is a group denoted by GL₂ and called the general linear group of order two.

2003, Maks Aizikovich Akivis, translated by Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 119:

We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups.

2009, Roe Goodman, Nolan R. Wallach, Symmetry, Representations, and Invariants, Springer, page 1:

(group theory) For given field F and order n, the group of invertible n×n matrices, with the group operation of matrix multiplication.


matrix group

The properties of matrix groups are of great importance for representation theory.

1976, S. Califano, Vibrational States, Wiley, page 112:

2011, László Babai, Finite Groups and Complexity Theory: From Leningrad to St Petersburg via Las Vegas, Alexander Kulikov, Nikolay Vereshchagin (editors), Computer Science – Theory and Applications: 6th International Computer Symposium, Springer, Lecture Notes in Computer Science 6651, page 162,
This paper is a personal account of the author's journey through the evolution of some of these interconnections, culminating in recent definitive results on the matrix group membership problem.

2014, Peter Sarnak, Notes on thin matrix groups, Emmanuel Breuillard, Hee Oh (editors), Thin Groups and Superstrong Approximation, Cambridge University Press, page 343,
Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given.

(group theory) Any group of invertible matrices over a specified field, with the group operation of matrix multiplication.


linear group

In this section we shall describe the linear groups and in the next two sections we deal with the symplectic and orthogonal groups, collectively usually known under the name of classical groups since the time of Hermann Weyl (1939).

2003, Paul M. Cohn, Further Algebra and Applications, Springer, page 116:

2007, Peter Schmid, The Solution of the k(GV) Problem, Imperial College Press, page vii,
So finite groups often appear as subgroups of permutation groups or linear groups. The “geometry” of a group (as permutation group or linear group, or as a group of Lie type etc.) should be used in order to describe basic invariants.

The standard examples are linear groups such as the orthogonal and unitary groups, or the additive groups of p-adic integers, and this confirms that the concept of a compact group is natural.

2007, Karl Heinrich Hofmann, Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups, European Mathematical Society, page 1:

(group theory) Any group that is isomorphic to a matrix group.


unitary group

The symmetric group is closely related to the unitary group.[…]The latest developments in spin free quantum chemistry have been the use of the unitary group for the calculation of the matrix elements of the Hamiltonian in the configuration interaction treatment.

1995, Ruben Pauncz, The Symmetric Group in Quantum Chemistry, CRC Press, page 195:

Two of the most important types of groups that occur in physics are unitary groups and orthogonal groups.

2004, [1999], Shoon K. Kim, Group Theoretical Methods and Applications to Molecules and Crystals, Cambridge University Press, page 53:

(group theory) For given n, the group of n×n unitary matrices, with the group operation of matrix multiplication.


orthogonal group

orthogonal group (plural orthogonal groups)

The symbol O^ε(n,q) for orthogonal groups has been well established in finite group theory as and, throughout the mathematics community, O(n, K) stands for an orthogonal group when K is the real or complex field.

1998, Robert L. Griess, Jr., Twelve Sporadic Groups, Springer, page 4:

Theorem 7.4. Let n ≥ 1. For odd primes p#x5C;not#x5C;equiv#x5C;pm 1#x5C;pmod#x7B;24#x7D; the odd-dimensional orthogonal groups O#x5F;#x7B;2n#x2B;1#x7D;(p) possess GA-realizations over #x5C;mathbb#x7B;Q#x7D;.

1999, Gunter Malle, B.H. Matzat, Inverse Galois Theory, Springer, page 146:

The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.

2007, Marcelo Epstein, Marek Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Springer, page 106:

(group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.


symplectic group

symplectic group (plural symplectic groups)

So named by German mathematician Hermann Weyl, replacing previous confusing names. More at symplectic.

This representation ω arises by virtue of the existence of an action by automorphisms of the symplectic group on a certain two-step nilpotent group, commonly called the Heisenberg group.
Weil was of course working in the adelic formalism of modern number theory, so he considered symplectic groups and Heisenberg groups with coefficients in a general local field.

1985, Roger Howe, “Dual Pairs in Physics: Harmonic Oscillators, Photons, Electrons, and Singletons”, in Mosh Flato, Paul Sally, Gregg Zuckerman, editors, Applications of Group Theory in Physics and Mathematical Physics, American Mathematical Society, page 179:

2001, G. Wassermann (translator), V. I. Arnol'd, A. B. Givental', Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 18,
The exponential of an operator gives the exponential mapping H↦exp(H)=∑Hᵏ/k! of the space of Hamiltonian operators to the symplectic group. The symplectic group acts by conjugation on itself and on its Lie algebra.

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group.

2012, Rolf Berndt, Ralf Schmidt, Elements of the Representation Theory of the Jacobi Group, Springer, page v:

(linear algebra, group theory) For given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.


חבורה

he-noun

חֲבוּרָה • (ḥavurá) f

חֲבוּרָה (canonical)
ḥavurá (romanization)


陪集

/pʰeɪ̯³⁵ t͡ɕi³⁵/

/pʰeɪ̯³⁵ t͡ɕi³⁵/


permutaatioryhmä

permutaatioryhmä (nominative singular)
permutaatioryhmät (nominative plural)
permutaatioryhmä (accusative nominative singular)
permutaatioryhmät (accusative nominative plural)
permutaatioryhmän (accusative genitive singular)
permutaatioryhmät (accusative genitive plural)
permutaatioryhmän (genitive singular)
permutaatioryhmien (genitive plural)
permutaatioryhmäin (genitive plural rare)
permutaatioryhmää (partitive singular)
permutaatioryhmiä (partitive plural)
permutaatioryhmässä (inessive singular)
permutaatioryhmissä (inessive plural)
permutaatioryhmästä (elative singular)
permutaatioryhmistä (elative plural)
permutaatioryhmään (illative singular)
permutaatioryhmiin (illative plural)
permutaatioryhmällä (adessive singular)
permutaatioryhmillä (adessive plural)
permutaatioryhmältä (ablative singular)
permutaatioryhmiltä (ablative plural)
permutaatioryhmälle (allative singular)
permutaatioryhmille (allative plural)
permutaatioryhmänä (essive singular)
permutaatioryhminä (essive plural)
permutaatioryhmäksi (singular translative)
permutaatioryhmiksi (plural translative)
permutaatioryhmättä (abessive singular)
permutaatioryhmittä (abessive plural)
permutaatioryhmin (instructive plural)
See the possessive forms below. (comitative)
permutaatioryhmäni (nominative singular)
permutaatioryhmäni (nominative plural)
permutaatioryhmäni (accusative nominative singular)
permutaatioryhmäni (accusative nominative plural)
permutaatioryhmäni (accusative genitive singular)
permutaatioryhmäni (accusative genitive plural)
permutaatioryhmäni (genitive singular)
permutaatioryhmieni (genitive plural)
permutaatioryhmäini (genitive plural rare)
permutaatioryhmääni (partitive singular)
permutaatioryhmiäni (partitive plural)
permutaatioryhmässäni (inessive singular)
permutaatioryhmissäni (inessive plural)
permutaatioryhmästäni (elative singular)
permutaatioryhmistäni (elative plural)
permutaatioryhmääni (illative singular)
permutaatioryhmiini (illative plural)
permutaatioryhmälläni (adessive singular)
permutaatioryhmilläni (adessive plural)
permutaatioryhmältäni (ablative singular)
permutaatioryhmiltäni (ablative plural)
permutaatioryhmälleni (allative singular)
permutaatioryhmilleni (allative plural)
permutaatioryhmänäni (essive singular)
permutaatioryhminäni (essive plural)
permutaatioryhmäkseni (singular translative)
permutaatioryhmikseni (plural translative)
permutaatioryhmättäni (abessive singular)
permutaatioryhmittäni (abessive plural)
permutaatioryhmineni (comitative plural)
permutaatioryhmäsi (nominative singular)
permutaatioryhmäsi (nominative plural)
permutaatioryhmäsi (accusative nominative singular)
permutaatioryhmäsi (accusative nominative plural)
permutaatioryhmäsi (accusative genitive singular)
permutaatioryhmäsi (accusative genitive plural)
permutaatioryhmäsi (genitive singular)
permutaatioryhmiesi (genitive plural)
permutaatioryhmäisi (genitive plural rare)
permutaatioryhmääsi (partitive singular)
permutaatioryhmiäsi (partitive plural)
permutaatioryhmässäsi (inessive singular)
permutaatioryhmissäsi (inessive plural)
permutaatioryhmästäsi (elative singular)
permutaatioryhmistäsi (elative plural)
permutaatioryhmääsi (illative singular)
permutaatioryhmiisi (illative plural)
permutaatioryhmälläsi (adessive singular)
permutaatioryhmilläsi (adessive plural)
permutaatioryhmältäsi (ablative singular)
permutaatioryhmiltäsi (ablative plural)
permutaatioryhmällesi (allative singular)
permutaatioryhmillesi (allative plural)
permutaatioryhmänäsi (essive singular)
permutaatioryhminäsi (essive plural)
permutaatioryhmäksesi (singular translative)
permutaatioryhmiksesi (plural translative)
permutaatioryhmättäsi (abessive singular)
permutaatioryhmittäsi (abessive plural)
permutaatioryhminesi (comitative plural)
permutaatioryhmämme (nominative singular)
permutaatioryhmämme (nominative plural)
permutaatioryhmämme (accusative nominative singular)
permutaatioryhmämme (accusative nominative plural)
permutaatioryhmämme (accusative genitive singular)
permutaatioryhmämme (accusative genitive plural)
permutaatioryhmämme (genitive singular)
permutaatioryhmiemme (genitive plural)
permutaatioryhmäimme (genitive plural rare)
permutaatioryhmäämme (partitive singular)
permutaatioryhmiämme (partitive plural)
permutaatioryhmässämme (inessive singular)
permutaatioryhmissämme (inessive plural)
permutaatioryhmästämme (elative singular)
permutaatioryhmistämme (elative plural)
permutaatioryhmäämme (illative singular)
permutaatioryhmiimme (illative plural)
permutaatioryhmällämme (adessive singular)
permutaatioryhmillämme (adessive plural)
permutaatioryhmältämme (ablative singular)
permutaatioryhmiltämme (ablative plural)
permutaatioryhmällemme (allative singular)
permutaatioryhmillemme (allative plural)
permutaatioryhmänämme (essive singular)
permutaatioryhminämme (essive plural)
permutaatioryhmäksemme (singular translative)
permutaatioryhmiksemme (plural translative)
permutaatioryhmättämme (abessive singular)
permutaatioryhmittämme (abessive plural)
permutaatioryhminemme (comitative plural)
permutaatioryhmänne (nominative singular)
permutaatioryhmänne (nominative plural)
permutaatioryhmänne (accusative nominative singular)
permutaatioryhmänne (accusative nominative plural)
permutaatioryhmänne (accusative genitive singular)
permutaatioryhmänne (accusative genitive plural)
permutaatioryhmänne (genitive singular)
permutaatioryhmienne (genitive plural)
permutaatioryhmäinne (genitive plural rare)
permutaatioryhmäänne (partitive singular)
permutaatioryhmiänne (partitive plural)
permutaatioryhmässänne (inessive singular)
permutaatioryhmissänne (inessive plural)
permutaatioryhmästänne (elative singular)
permutaatioryhmistänne (elative plural)
permutaatioryhmäänne (illative singular)
permutaatioryhmiinne (illative plural)
permutaatioryhmällänne (adessive singular)
permutaatioryhmillänne (adessive plural)
permutaatioryhmältänne (ablative singular)
permutaatioryhmiltänne (ablative plural)
permutaatioryhmällenne (allative singular)
permutaatioryhmillenne (allative plural)
permutaatioryhmänänne (essive singular)
permutaatioryhminänne (essive plural)
permutaatioryhmäksenne (singular translative)
permutaatioryhmiksenne (plural translative)
permutaatioryhmättänne (abessive singular)
permutaatioryhmittänne (abessive plural)
permutaatioryhminenne (comitative plural)
permutaatioryhmänsä (nominative singular)
permutaatioryhmänsä (nominative plural)
permutaatioryhmänsä (accusative nominative singular)
permutaatioryhmänsä (accusative nominative plural)
permutaatioryhmänsä (accusative genitive singular)
permutaatioryhmänsä (accusative genitive plural)
permutaatioryhmänsä (genitive singular)
permutaatioryhmiensä (genitive plural)
permutaatioryhmäinsä (genitive plural rare)
permutaatioryhmäänsä (partitive singular)
permutaatioryhmiään (partitive plural)
permutaatioryhmiänsä (partitive plural)
permutaatioryhmässään (inessive singular)
permutaatioryhmässänsä (inessive singular)
permutaatioryhmissään (inessive plural)
permutaatioryhmissänsä (inessive plural)
permutaatioryhmästään (elative singular)
permutaatioryhmästänsä (elative singular)
permutaatioryhmistään (elative plural)
permutaatioryhmistänsä (elative plural)
permutaatioryhmäänsä (illative singular)
permutaatioryhmiinsä (illative plural)
permutaatioryhmällään (adessive singular)
permutaatioryhmällänsä (adessive singular)
permutaatioryhmillään (adessive plural)
permutaatioryhmillänsä (adessive plural)
permutaatioryhmältään (ablative singular)
permutaatioryhmältänsä (ablative singular)
permutaatioryhmiltään (ablative plural)
permutaatioryhmiltänsä (ablative plural)
permutaatioryhmälleen (allative singular)
permutaatioryhmällensä (allative singular)
permutaatioryhmilleen (allative plural)
permutaatioryhmillensä (allative plural)
permutaatioryhmänään (essive singular)
permutaatioryhmänänsä (essive singular)
permutaatioryhminään (essive plural)
permutaatioryhminänsä (essive plural)
permutaatioryhmäkseen (singular translative)
permutaatioryhmäksensä (singular translative)
permutaatioryhmikseen (plural translative)
permutaatioryhmiksensä (plural translative)
permutaatioryhmättään (abessive singular)
permutaatioryhmättänsä (abessive singular)
permutaatioryhmittään (abessive plural)
permutaatioryhmittänsä (abessive plural)
permutaatioryhmineen (comitative plural)
permutaatioryhminensä (comitative plural)

(algebra, group theory) permutation group


tekijäryhmä

tekijäryhmä (nominative singular)
tekijäryhmät (nominative plural)
tekijäryhmä (accusative nominative singular)
tekijäryhmät (accusative nominative plural)
tekijäryhmän (accusative genitive singular)
tekijäryhmät (accusative genitive plural)
tekijäryhmän (genitive singular)
tekijäryhmien (genitive plural)
tekijäryhmäin (genitive plural rare)
tekijäryhmää (partitive singular)
tekijäryhmiä (partitive plural)
tekijäryhmässä (inessive singular)
tekijäryhmissä (inessive plural)
tekijäryhmästä (elative singular)
tekijäryhmistä (elative plural)
tekijäryhmään (illative singular)
tekijäryhmiin (illative plural)
tekijäryhmällä (adessive singular)
tekijäryhmillä (adessive plural)
tekijäryhmältä (ablative singular)
tekijäryhmiltä (ablative plural)
tekijäryhmälle (allative singular)
tekijäryhmille (allative plural)
tekijäryhmänä (essive singular)
tekijäryhminä (essive plural)
tekijäryhmäksi (singular translative)
tekijäryhmiksi (plural translative)
tekijäryhmättä (abessive singular)
tekijäryhmittä (abessive plural)
tekijäryhmin (instructive plural)
See the possessive forms below. (comitative)
tekijäryhmäni (nominative singular)
tekijäryhmäni (nominative plural)
tekijäryhmäni (accusative nominative singular)
tekijäryhmäni (accusative nominative plural)
tekijäryhmäni (accusative genitive singular)
tekijäryhmäni (accusative genitive plural)
tekijäryhmäni (genitive singular)
tekijäryhmieni (genitive plural)
tekijäryhmäini (genitive plural rare)
tekijäryhmääni (partitive singular)
tekijäryhmiäni (partitive plural)
tekijäryhmässäni (inessive singular)
tekijäryhmissäni (inessive plural)
tekijäryhmästäni (elative singular)
tekijäryhmistäni (elative plural)
tekijäryhmääni (illative singular)
tekijäryhmiini (illative plural)
tekijäryhmälläni (adessive singular)
tekijäryhmilläni (adessive plural)
tekijäryhmältäni (ablative singular)
tekijäryhmiltäni (ablative plural)
tekijäryhmälleni (allative singular)
tekijäryhmilleni (allative plural)
tekijäryhmänäni (essive singular)
tekijäryhminäni (essive plural)
tekijäryhmäkseni (singular translative)
tekijäryhmikseni (plural translative)
tekijäryhmättäni (abessive singular)
tekijäryhmittäni (abessive plural)
tekijäryhmineni (comitative plural)
tekijäryhmäsi (nominative singular)
tekijäryhmäsi (nominative plural)
tekijäryhmäsi (accusative nominative singular)
tekijäryhmäsi (accusative nominative plural)
tekijäryhmäsi (accusative genitive singular)
tekijäryhmäsi (accusative genitive plural)
tekijäryhmäsi (genitive singular)
tekijäryhmiesi (genitive plural)
tekijäryhmäisi (genitive plural rare)
tekijäryhmääsi (partitive singular)
tekijäryhmiäsi (partitive plural)
tekijäryhmässäsi (inessive singular)
tekijäryhmissäsi (inessive plural)
tekijäryhmästäsi (elative singular)
tekijäryhmistäsi (elative plural)
tekijäryhmääsi (illative singular)
tekijäryhmiisi (illative plural)
tekijäryhmälläsi (adessive singular)
tekijäryhmilläsi (adessive plural)
tekijäryhmältäsi (ablative singular)
tekijäryhmiltäsi (ablative plural)
tekijäryhmällesi (allative singular)
tekijäryhmillesi (allative plural)
tekijäryhmänäsi (essive singular)
tekijäryhminäsi (essive plural)
tekijäryhmäksesi (singular translative)
tekijäryhmiksesi (plural translative)
tekijäryhmättäsi (abessive singular)
tekijäryhmittäsi (abessive plural)
tekijäryhminesi (comitative plural)
tekijäryhmämme (nominative singular)
tekijäryhmämme (nominative plural)
tekijäryhmämme (accusative nominative singular)
tekijäryhmämme (accusative nominative plural)
tekijäryhmämme (accusative genitive singular)
tekijäryhmämme (accusative genitive plural)
tekijäryhmämme (genitive singular)
tekijäryhmiemme (genitive plural)
tekijäryhmäimme (genitive plural rare)
tekijäryhmäämme (partitive singular)
tekijäryhmiämme (partitive plural)
tekijäryhmässämme (inessive singular)
tekijäryhmissämme (inessive plural)
tekijäryhmästämme (elative singular)
tekijäryhmistämme (elative plural)
tekijäryhmäämme (illative singular)
tekijäryhmiimme (illative plural)
tekijäryhmällämme (adessive singular)
tekijäryhmillämme (adessive plural)
tekijäryhmältämme (ablative singular)
tekijäryhmiltämme (ablative plural)
tekijäryhmällemme (allative singular)
tekijäryhmillemme (allative plural)
tekijäryhmänämme (essive singular)
tekijäryhminämme (essive plural)
tekijäryhmäksemme (singular translative)
tekijäryhmiksemme (plural translative)
tekijäryhmättämme (abessive singular)
tekijäryhmittämme (abessive plural)
tekijäryhminemme (comitative plural)
tekijäryhmänne (nominative singular)
tekijäryhmänne (nominative plural)
tekijäryhmänne (accusative nominative singular)
tekijäryhmänne (accusative nominative plural)
tekijäryhmänne (accusative genitive singular)
tekijäryhmänne (accusative genitive plural)
tekijäryhmänne (genitive singular)
tekijäryhmienne (genitive plural)
tekijäryhmäinne (genitive plural rare)
tekijäryhmäänne (partitive singular)
tekijäryhmiänne (partitive plural)
tekijäryhmässänne (inessive singular)
tekijäryhmissänne (inessive plural)
tekijäryhmästänne (elative singular)
tekijäryhmistänne (elative plural)
tekijäryhmäänne (illative singular)
tekijäryhmiinne (illative plural)
tekijäryhmällänne (adessive singular)
tekijäryhmillänne (adessive plural)
tekijäryhmältänne (ablative singular)
tekijäryhmiltänne (ablative plural)
tekijäryhmällenne (allative singular)
tekijäryhmillenne (allative plural)
tekijäryhmänänne (essive singular)
tekijäryhminänne (essive plural)
tekijäryhmäksenne (singular translative)
tekijäryhmiksenne (plural translative)
tekijäryhmättänne (abessive singular)
tekijäryhmittänne (abessive plural)
tekijäryhminenne (comitative plural)
tekijäryhmänsä (nominative singular)
tekijäryhmänsä (nominative plural)
tekijäryhmänsä (accusative nominative singular)
tekijäryhmänsä (accusative nominative plural)
tekijäryhmänsä (accusative genitive singular)
tekijäryhmänsä (accusative genitive plural)
tekijäryhmänsä (genitive singular)
tekijäryhmiensä (genitive plural)
tekijäryhmäinsä (genitive plural rare)
tekijäryhmäänsä (partitive singular)
tekijäryhmiään (partitive plural)
tekijäryhmiänsä (partitive plural)
tekijäryhmässään (inessive singular)
tekijäryhmässänsä (inessive singular)
tekijäryhmissään (inessive plural)
tekijäryhmissänsä (inessive plural)
tekijäryhmästään (elative singular)
tekijäryhmästänsä (elative singular)
tekijäryhmistään (elative plural)
tekijäryhmistänsä (elative plural)
tekijäryhmäänsä (illative singular)
tekijäryhmiinsä (illative plural)
tekijäryhmällään (adessive singular)
tekijäryhmällänsä (adessive singular)
tekijäryhmillään (adessive plural)
tekijäryhmillänsä (adessive plural)
tekijäryhmältään (ablative singular)
tekijäryhmältänsä (ablative singular)
tekijäryhmiltään (ablative plural)
tekijäryhmiltänsä (ablative plural)
tekijäryhmälleen (allative singular)
tekijäryhmällensä (allative singular)
tekijäryhmilleen (allative plural)
tekijäryhmillensä (allative plural)
tekijäryhmänään (essive singular)
tekijäryhmänänsä (essive singular)
tekijäryhminään (essive plural)
tekijäryhminänsä (essive plural)
tekijäryhmäkseen (singular translative)
tekijäryhmäksensä (singular translative)
tekijäryhmikseen (plural translative)
tekijäryhmiksensä (plural translative)
tekijäryhmättään (abessive singular)
tekijäryhmättänsä (abessive singular)
tekijäryhmittään (abessive plural)
tekijäryhmittänsä (abessive plural)
tekijäryhmineen (comitative plural)
tekijäryhminensä (comitative plural)


primitive element

en-noun

primitive element (plural primitive elements)

primitiivinen alkio

element of a coalgebra satisfying a particular condition

In this paper we apply regression models and other pattern recognition techniques to the task of classifying primitive elements of a free group.

2004, Dmitry Y. Bormotov, “Experimenting with Primitive Elements in F₂”, in Alexandre Borovik, Alexei G. Myasnikov, editors, Computational and Experimental Group Theory: AMS-ASL Joint Special Session, American Mathematical Society, page 215:

(group theory, of a free group) An element of a free generating set of a given free group.


automorphism group

automorphism group (plural automorphism groups)

The automorphism group of a set is the set's symmetric group.

The automorphism group of a finite-dimensional vector space (respectively, projective space) is the space's general linear group (respectively, projective linear group).

Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds.

1984, José M. Isidro, László L. Stachó, Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, Elsevier (North-Holland), page vi:

Recently Droste, Holland and Macpherson in [DHM1], [DHM2] and [DHM3] investigated various questions concerning the automorphism groups of homogeneous #x5C;aleph#x5F;0-categorical trees.

1993, Matatyahu Rubin, The Reconstruction of Trees from Their Automorphism Groups, American Mathematical Society, page 5:

Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups.

2002, Masamichi Takesaki, Theory of Operator Algebras II, Springer, page 91:

function composition
function composition

(group theory) The group of automorphisms of a given set or other mathematical object, with the group operation being function composition.


lamplighter group

lamplighter group (plural lamplighter groups)

The group can be viewed as acting on a doubly infinite sequence of street lamps …,l_-2,l_-1,l_0,l_1,l_2,l_3,… each of which may be on or off, with a lamplighter standing at some lamp l_k.

(group theory) The restricted wreath product mathbf Z₂≀ mathbf Z.


mříž

mříž (nominative singular)
mříže (nominative plural)
mříže (genitive singular)
mříží (genitive plural)
mříži (dative singular)
mřížím (dative plural)
mříž (accusative singular)
mříže (accusative plural)
mříži (singular vocative)
mříže (plural vocative)
mříži (locative singular)
mřížích (locative plural)
mříží (instrumental singular)
mřížemi (instrumental plural)

Inherited from Old Czech mřiežě, from Proto-Slavic *merža.


cyclic number

Euler's totient function
Euler's totient function

(group theory) A natural number n such that n and φ(n) (that is, Euler's totient function) are coprime.


complexity function

complexity function (plural complexity functions)

Komplexitätsfunktion

function that counts distinct factors of a string

It follows that
#x5C;bullet The length function l#x3A;A#x2A;#x5C;longrightarrow#x5C;mathcalN is a complexity function on A#x2A;.

2002, A. V. Borovnik, A. G. Myasnikov, V. Shpilrain, “Measuring Sets in Infinite Groups”, in Robert H. Gilman, Alexei G. Myasnikov, Vladimir Shpilrain, editors, Computational and Statistical Group Theory, American Mathematical Society, page 27:

2003, Julien Cassaigne, Constructing Infinite Words of Intermediate Complexity, Masami Ito, Masafumi Toyama (editors), Developments in Language Theory: 6th International Conference, 6th International Conference, DLT 2002, Revised Papers, Springer, LNCS 2450, page 173,
We present two constructions of infinite words with a complexity function that grows faster than any polynomial, but slower than any exponential.

(group theory, computing theory, of a string) A function that counts the number of distinct factors (substrings of consecutive symbols) in a string of symbols;

A function that counts the number of distinct factors (substrings of consecutive symbols) in a string of symbols;
(of a formal language) a function that counts the number of words of a given length.

Komplexitätsfunktion

function that counts distinct words of a given length in a language


Tits group

exceptional isomorphism
exceptional isomorphism

(group theory) A finite simple group of order 2¹¹ · 3³ · 5² · 13 = 17,971,200. It is the only simple group that is a derivative of a group of Lie type that is not strictly a group of Lie type in any series due to exceptional isomorphism.


group of Lie type

group of Lie type (plural groups of Lie type)

Named for their close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers.

(group theory) Usually, a finite group that is closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.


reduced word

(group theory) A word that contains no redundant pairs.