Let $H$ be a subgroup of $G$, and $x \in N\_G(H)$. Prove that $$P = \{hx^i | h\in H, i\in\Bbb Z\}$$ is a subgroup of $G$.

I proved that $P$ is not empty because $H$ is a subgroup of $G$ so it's not empty and $N\_G(H)$ is also not empty because $1$ commutes with every element in $H$.

I don't know how to show that $P$ is closed under $G$ operation and is closed under inverses.

To prove the former, I tried: Let $s$ and $t \in P$ so $s = h\_1(x\_1)^i$ and $t = h\_2(x\_2)^j$, so $st = h\_1(x\_1)^ih\_2(x\_2)^j$ but don't know how to show that belongs to $P$ also...

I would really appreciate any help.