Thread: Preserving structural properties

1. Preserving structural properties

Prove that if G′ is cyclic then so is G.

Thanks for the help!

2. Let $f(a)=b \in G'$ be a generator for $G'$ with $a \in G$. Since every $y \in G'$ is of the form $b^n$ for some $n$ we have $y=b^n=f(a)^n=f(a^n)$ ie. every element in the image of f comes from some $a^n$, since $f$ is one-one we conclude that every element in $G$ is of the form $a^n$ which means $G$ is cyclic