If group homomorphisms preserve inverses, why aren't they just isomorphisms?

Question

If group homomorphisms preserve inverses, why aren't they just isomorphisms? Because if the homomorphism $\phi$ preserves inverses, then isn't $\phi$ bijective hence an isomorphism?

Answer 1

No: for $\phi$ to be bijective it needs to have an inverse as a map, which is totally different from preserving inverses in the group structure. To have an inverse would mean there is a map $\phi^{-1}$ such that the compositions $\phi\circ\phi^{-1}$ and $\phi^{-1}\circ\phi$ are the identity maps. To preserve inverses means that for any $g$ in the domain of $\phi$, $\phi(g^{-1})=\phi(g)^{-1}$.

For a simple concrete example, consider the homomorphism $\phi:\mathbb{Z}\to\mathbb{Z}$ given by $\phi(n)=0$ for all $n$. This map preserves inverses, meaning that $\phi(-n)=-\phi(n)$ for all $n$ (both sides are equal to $0$). But it certainly is not a bijection!