Testing Normality in a Group

Question

Let H be a normal subgroup of G. Assume that $ab \in H$. Show $ba\in H$. Consider the conjugate of $ab\in H$ with the element $b$.

My question is what is the conjugate of a $ab$ with $b$?

Also the second part of the question is, If H is a subgroup of G and satisfies the property that $ab \in H$ implies $ba\in H$ show that H is normal in G.

I think the second one should be a little bit easier once the first part has been solved, maybe I will see something, but some help would nice on this part too.

Answer 1

Definition: two elements $\;x,y\;$ in a group $\;G\;$ are called conjugate if there exists $\;g\in G\;$ s.t. $\;y=g^{-1}xg\;$

$$(1)\;\;ba=a^{-1}\left(ab\right)a$$

Now, if $\;ab\in H\iff ba\in H\;$ , then for $\;h\in H\;,\;\;x\in G\;$ :

$$x^{-1}hx=x^{-1}\left(hx\right)\in H\iff(hx)x^{-1}\in H$$

and since the last containtment is trivial we get that $\;x^{-1}hx\in H\implies H\lhd G\;$