Does any homomorphism map generators to generators?

Question

I think similar questions may have been asked, so apologies if this is the case. I haven't been able to find one that helps me yet though.

Anyway, for an assignment question I've figured out a solution but it relies on my assumption that an isomorphism between cyclic groups maps generators to generators. I've seen this stated before but I'm not quite sure how to prove it.

Could anyone help me? I attempted using the idea that the identity gets mapped to the identity and trying to generalise to generators but got stuck.

Thanks in advance!

Answer 1

If $g$ generates $G$, then the elements of $G$ are precisely $g,g^2,\ldots,g^n$ where $n$ is the order of $g$.

If $\phi : G \to G'$ is an isomorphism, then consider where an arbitrary element of $G$ is sent. An arbitrary element of $G$ is of the form $g^k$, so $\phi(g^k)=\phi(g)^k$. Since $\phi$ is a bijection, the elements of $G'$ are precisely $\phi(g),\phi(g)^2,\ldots,\phi(g)^n$.

Answer 2

No. Consider the trivial homomorphism $\phi: Z_n\to Z_m$ that takes $g\in Z_n$ to $e\in Z_m$. $e$ isn't a generator of $Z_m$, but $\phi(g_1g_2)=e$ and $\phi(g_1)\phi(g_2)=e^2=e$, so $\phi$ is a homomorphism.