Direct factor is not intersection-closed

This article gives the statement, and possibly proof, of a subgroup property not satisfying a subgroup metaproperty .
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Statement

The intersection of two direct factors of a group need not be a direct factor. In fact, this property is not even true within $p$ -groups.

Proof

The counterexample

Let $C_{n}$ denote the cyclic group on $n$ elements. Then, consider the group $C_{p} \times C_{p^{2}}$ . Consider the automorphism $σ$ of this group which sends the pair $(a, b)$ to $(a + b, b)$ . Consider the intersection of the group $C_{p^{2}}$ (which is a direct factor) with its image under $σ$ . This is basically the subgroup of $C_{p^{2}}$ comprising elements which are multiples of $p$ . Clearly, this is not a direct factor, in fact it is not even a direct factor inside $C_{p^{2}}$ .