Let H is a normal subgroup of G.
For any $x\in H$, $x^H=\{h^{-1}xh:h\in H\}$ .
A normal subgroup H of a group G is said to be conjugate closed, if $x^G=x^H$ for $x\in H$.
A group G is said to be conjugate closed if every normal subgroup of G is conjugate closed.
Prove that: A direct product of conjugate closed groups is conjugate closed group.
i.e.: If $G_1$ and $G_2$ is conjugate closed group, $G=G_1\times G_2$ is conjugate closed group!
Of course, we have $x^H\subset x^G$, so what $x^G\subset x^H$?
I try to prove it, but i can't. Please help me!