Coset

English

Etymology

From Template:Prefix; apparently first used 1910 by American mathematician Template:W.

Pronunciation

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Noun

1. Template:Lb 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.
   • 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, Template:W, 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.

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Usage notes

Mathematically, given a group $G$ with binary operation $\circ$, element $g\in G$ and subgroup $H\subseteq G$, the set $\{g\circ h:h\in H\}$, which also defines the Template:M if $G$ is not assumed to be abelian.

The concept is relevant to the (mathematical) definitions of Template:M and Template:M.

Translations

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Further reading

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• Coset on Template:W
• Coset in a group on Template:W

Anagrams

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