coset

Etymology

 * apparently first used 1910 by American mathematician.

Noun

 * 1)  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.
 * 2) * 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.
 * 1) * 1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics,, 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.

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

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

Translations

 * Estonian: kõrvalklass
 * Finnish: sivuluokka
 * Italian: classe laterale
 * Polish:
 * Portuguese: classe lateral
 * Swedish: sidoklass