See coset in All languages combined, or Wiktionary
Download JSON data for coset meaning in English (4.4kB)
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"etymology_text": "co- + set; apparently first used 1910 by American mathematician George Abram Miller.",
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"form": "cosets",
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"name": "Algebra",
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"word": "double coset"
},
{
"word": "left coset"
},
{
"word": "right coset"
}
],
"examples": [
{
"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."
},
{
"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;.",
"type": "quotation"
}
],
"glosses": [
"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."
],
"id": "en-coset-en-noun-Acvql~mj",
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"raw_glosses": [
"(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."
],
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"algebra",
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"mathematics",
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"translations": [
{
"code": "et",
"lang": "Estonian",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"word": "kõrvalklass"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"word": "sivuluokka"
},
{
"code": "it",
"lang": "Italian",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "classe laterale"
},
{
"code": "pl",
"lang": "Polish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "warstwa"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "classe lateral"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"common-gender"
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"word": "sidoklass"
}
],
"wikipedia": [
"George Abram Miller",
"coset"
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}
],
"word": "coset"
}
{
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{
"word": "right coset"
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"examples": [
{
"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."
},
{
"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;.",
"type": "quotation"
}
],
"glosses": [
"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."
],
"links": [
[
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"algebra"
],
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"(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."
],
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"translations": [
{
"code": "et",
"lang": "Estonian",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"word": "kõrvalklass"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"word": "sivuluokka"
},
{
"code": "it",
"lang": "Italian",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "classe laterale"
},
{
"code": "pl",
"lang": "Polish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "warstwa"
},
{
"code": "pt",
"lang": "Portuguese",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"feminine"
],
"word": "classe lateral"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "result of applying group operation with fixed element from the parent group on each element of a subgroup",
"tags": [
"common-gender"
],
"word": "sidoklass"
}
],
"word": "coset"
}
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