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

Thanks for the help!

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- Oct 22nd 2009, 03:57 PMKiwili49Preserving structural properties
Prove that if G′ is cyclic then so is G.

Thanks for the help! - Oct 22nd 2009, 04:36 PMJose27
Let $\displaystyle f(a)=b \in G'$ be a generator for $\displaystyle G'$ with $\displaystyle a \in G$. Since every $\displaystyle y \in G'$ is of the form $\displaystyle b^n$ for some $\displaystyle n$ we have $\displaystyle y=b^n=f(a)^n=f(a^n)$ ie. every element in the image of f comes from some $\displaystyle a^n$, since $\displaystyle f$ is one-one we conclude that every element in $\displaystyle G$ is of the form $\displaystyle a^n$ which means $\displaystyle G$ is cyclic