Prove that $x^{−1}Hx $ is a subgroup of G Let $H$ be a subgroup of a group $G$ and, for $x\in G$, let $x^{-1}ax$ denote the set $\{x^{−1}ax : a\in H\}$. Prove that $x^{−1}Hx$ is a subgroup of $G$.

First of all, $1\in H^x=x^{-1}Hx$ and so $x^{-1}Hx$ is not empty. Let $x^{-1}h_1x,x^{-1}h_2x\in x^{-1}Hx$. Then $x^{-1}h_1h_2x\in x^{-1}Hx$. Furthermore, $(x^{-1}h_1x)^{-1}= x^{-1}{h_1}^{-1}x\in x^{-1}h_1x$.
