For a subgroup H of a group G and an element x of G, define xH to be the set {xh:h in H} and Hx to be the set {hx:h in H}. A subset of G of the form xH for some x in G is said to be a left coset of H and a subset of the form Hx is said to be a right coset of H.

For any subgroup H, we can define an equivalence relation ∼ by x∼y if x=yh for some h in H. The equivalence classes of this equivalence relation are exactly the left cosets of H, and an element x of G is in the equivalence class xH. Thus the left cosets of H form a partition of G.

It is also true that any two left cosets of H have the same cardinal number, and in particular, every coset of H has the same cardinal number as eH=H, where e is the identity element. Thus, the cardinal number of any left coset of H has cardinal number the order of H.

The same results are true of the right cosets of G and, in fact, one can prove that the set of left cosets of H has the same cardinal number as the set of right cosets of H.

See also

Coset Space, Equivalence Class, Group, Left Coset, Quotient Group, Right Coset, Subgroup

This entry contributed by Nicolas Bray

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Bray, Nicolas. "Coset." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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