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

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.

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.