See coset on Wiktionary
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"word": "double coset"
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"word": "left coset"
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"word": "multicoset"
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"text": "1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597,\nTheorem 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."
},
{
"text": "1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,\nIn general, the coset in row x consists of all the elements xh as h runs through the various elements of H."
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"ref": "2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, 3rd edition, Springer, page 231:",
"text": "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;.",
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"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."
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"id": "en-coset-en-noun-Acvql~mj",
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"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"word": "kõrvalklass"
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"text": "1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,\nIn general, the coset in row x consists of all the elements xh as h runs through the various elements of H."
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"text": "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;.",
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"word": "coset"
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This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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