Prove isomorphism of groups $\langle G, + , {}^{-1}\rangle$ and $\langle G, *,{}^{-1}\rangle$, where $a*b=b+a$
$\forall a,b \in G$
I'm barely starting to study abstract algebra.
So how do I show isomorphism? I think that I should show a homomorphism somehow, but I don't know how.
Any thoughts/ideas would be really appreciated!
On
Let $\phi (g)=g^{-1}\,,\forall g\in G$.
$\phi$ is a homomorphism: $\phi (g*h)=(g*h)^{-1}=h^{-1}*g^{-1}=g^{-1}+h^{-1}=\phi(g)+\phi(h)$.
The kernel of $\phi$ is trivial: $\phi(g)=e\implies g^{-1}=e\implies g=e$.
$\phi$ is surjective: $\forall g\in G,\phi(g^{-1})=g$.
Thus $\phi$ is an isomorphism.
Hint: Define $f(a)=a^{-1}$ and prove that it is an isomorphism.