TITLE: Product of right cosets equals right coset implies normality of subgroup
QUESTION [11 upvotes]: I cannot see how to find a way to prove that if $H$ is a subgroup of $G$ such that the product of two right cosets of $H$ is also a right coset of $H,$ then $H$ is normal in $G.$ 
(This is from Herstein by the way.)
Thank you.

REPLY [2 votes]: Suppose g ∈ G and consider the product of two right cosets HgHg−1
. Observe
egeg−1 = e ∈ HgHg−1 and since by hypothesis HgHg−1
is a right coset, it would have
to be the right coset H = He as right cosets are either equal or disjoint and in this case
e ∈ H = He so we must have equality HgHg−1 = H. Therefore, for every g ∈ G we have
HgHg−1 = H so clearly gHg−1⊆ HgHg−1 = H. Hence, gHg−1 ⊆ H for all g ∈ G so H
is a normal subgroup of G.