Socle is strictly characteristic
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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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.
- Monolith is strictly characteristic: This is a special case where there is a unique minimal normal subgroup, and this is contained in every nontrivial normal subgroup.
- Baer norm is strictly characteristic
- Fitting subgroup is strictly characteristic
- Solvable core is strictly characteristic
- Center is strictly characteristic
- Socle is normality-preserving endomorphism-invariant
- Normality-preserving endomorphism-invariant implies strictly characteristic
The proof follows directly from facts (1) and (2).