Socle is strictly characteristic

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

The socle of a group (defined as the subgroup generated by all minimal normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the socle to within itself.

Related facts

Facts used

  1. Socle is normality-preserving endomorphism-invariant
  2. Normality-preserving endomorphism-invariant implies strictly characteristic

Proof

The proof follows directly from facts (1) and (2).