cosetn.

Brit.

/ˈkəʊsɛt/, U.S.

/ˈkoʊˌsɛt/

Forms: Also co-set.

Etymology: < co- prefix 6 + set n.²

Mathematics.

A set composed of all the products obtained by multiplying, on the right or on the left, each element of a subgroup in turn by one particular element of the group containing the subgroup.

ΘΚΠ

the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > in abstract algebra

coset1910

superset1917

neighbourhood1934

orbit1939

support1952

1910 G. A. Miller in Q. Jrnl. Math. XLI. 382 If H represents any sub-group of the group G, all the operators of G may be represented..in either of the following two ways: G = H + HS₂ + HS₃ + … +HS_γ = H + T₂H + T₃H + … +T_γH. If for each co-set HS_α (α = 2, 3, …, γ) it is possible to find some co-set T_βH such that the totality of the operators HS_α coincide with that of T_βH, then H is an invariant sub-group of G.

1911 G. A. Miller in Trans. Amer. Math. Soc. 12 326 The sets HS_α, S_αH (α = 2, 3, …, ρ) have been called Nebengruppen or co-sets of G as regards H, the former being called the right co-sets and the latter the left co-sets.

1941 G. Birkhoff & S. MacLane Surv. Mod. Algebra vi. 146 We now come to a far-reaching concept of abstract group theory: the idea that any subgroup S of a group G decomposes G into cosets.

1973 W. Ledermann Introd. Group Theory ii. 33 The cosets Hx and Hy are identical if and only if xy⁻¹ ∈ H. Any two cosets are either identical or else have no element in common.

1984 C. C. Sims Abstract Algebra iii. 99 Let f : G → H be a homomorphism of groups. Then the kernel N of f is a normal subgroup of G, and two elements of G are mapped to the same element of H if and only if they are in the same coset of N.

This entry has not yet been fully updated (first published 1989; most recently modified version published online March 2022).

单词	coset
释义	cosetn. Brit. /ˈkəʊsɛt/, U.S. /ˈkoʊˌsɛt/ Forms: Also co-set. Etymology: < co- prefix 6 + set n.² Mathematics. A set composed of all the products obtained by multiplying, on the right or on the left, each element of a subgroup in turn by one particular element of the group containing the subgroup. ΘΚΠ the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > in abstract algebra coset1910 superset1917 neighbourhood1934 orbit1939 support1952 1910 G. A. Miller in Q. Jrnl. Math. XLI. 382 If H represents any sub-group of the group G, all the operators of G may be represented..in either of the following two ways: G = H + HS₂ + HS₃ + … +HS_γ = H + T₂H + T₃H + … +T_γH. If for each co-set HS_α (α = 2, 3, …, γ) it is possible to find some co-set T_βH such that the totality of the operators HS_α coincide with that of T_βH, then H is an invariant sub-group of G. 1911 G. A. Miller in Trans. Amer. Math. Soc. 12 326 The sets HS_α, S_αH (α = 2, 3, …, ρ) have been called Nebengruppen or co-sets of G as regards H, the former being called the right co-sets and the latter the left co-sets. 1941 G. Birkhoff & S. MacLane Surv. Mod. Algebra vi. 146 We now come to a far-reaching concept of abstract group theory: the idea that any subgroup S of a group G decomposes G into cosets. 1973 W. Ledermann Introd. Group Theory ii. 33 The cosets Hx and Hy are identical if and only if xy⁻¹ ∈ H. Any two cosets are either identical or else have no element in common. 1984 C. C. Sims Abstract Algebra iii. 99 Let f : G → H be a homomorphism of groups. Then the kernel N of f is a normal subgroup of G, and two elements of G are mapped to the same element of H if and only if they are in the same coset of N. This entry has not yet been fully updated (first published 1989; most recently modified version published online March 2022). < n.1910
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