If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup
Problem 469
Let $H$ be a subgroup of a group $G$.
Suppose that for each element $x\in G$, we have $x^2\in H$.
Then prove that $H$ is a normal subgroup of $G$.
(Purdue University, Abstract Algebra Qualifying Exam)
Add to solve later 
