If $G$ is a cyclic group of order $n$ and $p|n$, prove that there exists a homomorphism of $G$ onto a cyclic group of order $p$. Find its Kernel.

Question

If $G$ is a cyclic group of order $n$ and $p|n$, prove that there exists a homomorphism of $G$ onto a cyclic group of order $p$. Find its Kernel.

My try:

Since $G$ is cyclic and $p|n$ there is a unique subgroup of order $p$ in $G$. So let $G=\langle a\rangle$ and $|G|=n$ and $H=\langle a^m\rangle$ for some $m$ and $|H|=p$. We have to prove that $f:G\to H$ is a homomorphism. For that we need to show that it is $one-one$,$onto$ and $operation-preserving$. How to start? Any hints??

Answer 1

Set $m=n/p$ ($m$ is integer by assumption). Consider the map $$ \varphi\colon G\to G \qquad \varphi(x)=x^m $$ Show

1. $\varphi$ is a homomorphism
2. if $g$ is a generator of $G$, then $\varphi(g)\ne1$
3. $\varphi(G)=\langle g^m\rangle$ is cyclic

Can you finish?