Hi.

I need to prove:

$\displaystyle H\triangleleft G \Leftrightarrow \forall g\in G,\hspace{5} \forall h\in H:\hspace{5} g^{-1}hg\in H$

So, one direction is quite easy:

$\displaystyle \Rightarrow$ :

If $\displaystyle H$ is a subgroup of $\displaystyle G$, it follows that $\displaystyle \forall g\in G$: $\displaystyle gH=Hg$.

in particular, for every $\displaystyle h\in H$, $\displaystyle gh=hg$ , so $\displaystyle h=g^{-1}hg$, $\displaystyle \forall g\in G, \forall h\in H$.

but the second direction is not quite clear for me:

$\displaystyle \Leftarrow$ :

Since $\displaystyle \forall g\in G$, $\displaystyle \forall h\in H$: $\displaystyle g^{-1}hg\in H$, there exist some $\displaystyle h_0\in H$, such that $\displaystyle g^{-1}hg=h_0$

and then:

$\displaystyle gh_0=hg$

but how do I know that $\displaystyle h_0=h$?

Thanks in advance.