TITLE: on finite abelian groups
QUESTION [8 upvotes]: 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 $M(G)=1$.

REPLY [4 votes]: 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.