coset

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

gH = { gh : h an element of H } is the left coset of H in G with respect to g, and

Hg = { hg : h an element of H } is the right coset of H in G with respect to g.

Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup. Although derived from a subgroup, cosets are not usually themselves subgroups of G, only subsets.