# Let $\phi:\Bbb Z_n\rightarrow G$ s.t. $\phi(i)=h^i$ for $0\le i\le n$. Give necessary and sufficient condition for $\phi$ to be homomorphism.

Let $$G$$ be a group, $$h$$ and element of $$G$$, and $$n$$ a positive integer. Let $$\phi : \mathbb{Z}_n\rightarrow G$$ be defined by $$\phi(i)=h^i$$ for $$0\leq i\leq n$$. Give a necessary and sufficient condition (in terms of $$h$$ and $$n$$) for $$\phi$$ to be a homomorphism. Prove your assertion.

I always have problems with necessary and sufficient arguments, because I do not know how to prove. I know it implies an if and only if, but I'm not sure exactly over which is used. Now, if I look at the answer, it says:

The map is a homomorphism if and only if $$h^n=e$$, the identity in $$G$$.

But how do you know that I have to prove $$h^n=e$$?

I would have thought that I have to prove $$\phi(n+m)=\phi(n)\phi(m)$$, so it's quite confusing to think that we have to prove $$h^n=e$$.

• Could it be that the question is asking about $\phi:\Bbb Z_n \to G$? Commented Dec 21, 2017 at 18:27
• Missing an $n$ in $\mathbb{Z}$. Commented Dec 21, 2017 at 18:30
• Obvious for you. Commented Dec 21, 2017 at 18:40
• So maybe start by proving that $\tilde{\phi}: \mathbb{Z}\rightarrow G$, $i \mapsto h^i$ is always a homomorphism. Commented Dec 21, 2017 at 19:27
• After that, think about a condition on $\tilde{\phi}$ that would allow you to replace $\mathbb{Z}$ with $\mathbb{Z}/n\mathbb{Z}$ (hint: first isomorphism theorem). Commented Dec 21, 2017 at 19:31

1) Prove $$\tilde{\phi}: \Bbb Z\to G, i\mapsto h^i$$ is always a homomorphism.
2) Think of the condition on $$\tilde{\phi}$$ that would allow $$\Bbb Z$$ to be replaced by $$\Bbb Z/n\Bbb Z$$.