On a classification of all the characteristic subgroups of a finite abelian $p$-group.

Question

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is of the form $p^\alpha P[p^\beta]$.

I'd like to determine all the characteristic subgroups of a finite abelian $p$-group and I came to the above conjecture. But I don't know how to prove it. The facts I have proved are that $\text{Aut}(P)$ is transitive on the set of all the elements of highest order, and that for any $p^\alpha\mid p^\beta\mid\exp P$, $p^\alpha(P[p^\beta])=p^\alpha P[p^{\beta-\alpha}]$.

Can anyone help me with this? Thanks in advance.

Answer 1

It doesn't hold even for invariant subgroups. For example, let $P=C_{p^5}\times C_{p^3}\times C_p$, then $pP+P[p]$ is a invariant subgroup, but is not of the form that you described.

Answer 2

It's not true for $P=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z}$, is it?

The subgroup $Q$ generated by $(1,2)$ is characteristic (it's the only cyclic $Q$ of order $4$ such that $P/Q$ is also cyclic), but it's not of the form you describe.

Although it's not immediately obvious how to give a similar example for odd $p$.