$H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$

Question

Let $G$ be a group . If $H :=\{ g^2 : g \in G\}$ is a subgroup of $G$ , the how to prove that $H$ is normal in $G$ ?

Answer 1

Consider if $h\in G$ then $h^2 \in H$, so for any $g\in G$ we have:

$$gh^2g^{-1} = (ghg^{-1})^2 \in H$$

Answer 2

Hint: What is $(hgh^{-1})^n$ equal to, for any $n$?