Element of order $p$ and finite height $\geq1$ in a reduced abelian group $p$-group with an element of order $p^2$

Question

This is a reference request for the following statement:

Fact: Let $G$ be a reduced abelian $p$-group with an element of order $p^2$. Then $G$ contains an element of order $p$ and of finite height at least $1$.

The proof of this uses only basics properties of the height function $h_p(-)$ (and the result is trivial if $G$ has no elements of infinite height). However, instead of writing down the proof, I would prefer to make a reference to some textbook or paper, as I'm sure that this fact is well-known to specialists of abelian groups.

Answer 1

This might be using a sledgehammer to crack a nut, but a theorem of Kaplansky quoted on p. 200 of L. Fuchs, Abelian Groups, Springer, 2015 states:

Let $A$ be an abelian $p$--group of length $\tau$, $\sigma_0 < \sigma_1 < \cdots < \sigma_{n-1} < \tau$ a sequence of ordinals, then $(\sigma_0, \dots, \sigma_{n-1}, \sigma_n = \infty)$ is the indicator of some $a\in A$ of order $p^n$ if and only if it satisfies the gap condition: if $\sigma_{i+1} > \sigma_i + 1$ then the $\sigma_i$ Ulm--Kaplansky invariant of $A$ is non--zewro.