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.