Hello

. Need some fast help on this:

Let $\displaystyle \phi

G, \star)\rightarrow(H, \triangle) $ be an isomorphism between these two groups. If $\displaystyle a' \in G$ is the inverse element of $\displaystyle a \in G$, then prove that $\displaystyle \phi(a')\in H$ is the inverse element of $\displaystyle \phi(a) \in H$.

I tried like this:

Since $\displaystyle \phi$ is isomorphism, that means it is a one to one map thus

$\displaystyle \phi(a)=\phi(b)$ only when $\displaystyle a = b$ and since $\displaystyle \phi $ is isomorphism, $\displaystyle \phi(a')$ must be the inverse of $\displaystyle \phi(a)$.

I can't see any explanation or justification here...in fact, the argument is true for ANY group homomorphism, isomorphism or not, and the proof is boringly simple: $\displaystyle \phi(a)\phi(a')=\phi(aa')=\phi(e_G)=e_H$ Tonio
Do you think my prove is enough, if not please elaborate.

Thanks for your time.