Coset

In mathematics, if G is a group, H a subgroup of G, and g an element of G, then

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:gH = { gh : h an element of H } is a left coset of H in G, and

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:Hg = { hg : h an element of H } is a right coset of H in G.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We have gH = H if and only if g is an element of H.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Any two left cosets are either identical or disjoint. The left cosets form a partition of G: every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and H itself is the only coset that is a subgroup.

Group: The term group can refer to several concepts:...

Subgroup: In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation on H....

Disjoint: Disjoint may refer to:...