Socle is powering-invariant in solvable group

This article gives the statement, and possibly proof, of the fact that in any solvable group, the subgroup obtained by applying a given subgroup-defining function (i.e., socle) always satisfies a particular subgroup property (i.e., powering-invariant subgroup)
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Statement

Suppose $G$ is a solvable group. Then, the socle $\operatorname {Soc} (G)$, defined as the subgroup generated by all the minimal normal subgroups of $G$, is a powering-invariant subgroup of $G$.

Facts used

1. Minimal normal implies powering-invariant in solvable group

Proof

The proof follows from Fact (1), and the observation that since all the minimal normal subgroups pairwise centralize each other, their join, the socle, is also powering-invariant.