8
$\begingroup$

Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove

$$M(G)=\langle1\rangle \text{ or } Z_{2}$$

Attempt: We know that $M(G)$ is elementary abelian $2$-group.

Also if $G$ is an abelian group of finite odd order, then $M(G)=1$.

$\endgroup$
  • 2
    $\begingroup$ So you can assume $G$ is a 2-group and a direct product of cyclic groups. Note that $|M(G)|=2$ if and only if there is unique cyclic direct factor of maximal order. $\endgroup$ – Derek Holt Apr 12 '13 at 8:18
4
$\begingroup$

It is a good idea to prove this for the cyclic groups $\mathbb Z / p^n \mathbb Z$, then to use the decomposition of finite abelian groups.

$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.