Is it true that cyclic subgroups are always normal?

$\begingroup$

There is a proposition that I'm supposed to "prove", but it doesn't sound true to me. It says that if $H$ is a cyclic subgroup of a group $G$ (notation $H<G$), then every $K <H$ is normal in $G$.

If that were the case, since $H<H$, we'd have a corollary: If $H$ is cyclic, then $H$ is normal in $G$. But is that even true?

$\endgroup$ 4

1 Answer

$\begingroup$

No, it's not true that if $H$ is a cyclic subgroup of $G$ then it is a normal subgroup of $G$. For a simple counterexample, let $G=S_3$ and let $H$ be the subgroup generated by the transposition $(12)$.

Perhaps the problem should instead read "every $K\leq H$ is normal in $H$".

$\endgroup$ 1

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy