Suppose $\displaystyle G$ is a group, and $\displaystyle H$ is somesubsetof $\displaystyle G$.

Is it true that if $\displaystyle \forall g\in G$ and $\displaystyle \forall h\in H$: $\displaystyle ghg^{-1}\in H$, then $\displaystyle H$ is anormal subgroup?

Or that $\displaystyle H$ has to be asubgroup(not just any subset) for this statment to be true?

I'm asking because I need to prove it, and I not 100 percent sure my teacher didn't make a mistake here...

Thanks in advance.