Fitting subgroup 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., Fitting subgroup) always satisfies a particular subgroup property (i.e., strictly characteristic subgroup)}
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Statement

The Fitting subgroup of a group (i.e., the subgroup generated by all nilpotent normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends it to within itself.

Related facts

Similar results include:

Facts used

  1. Fitting subgroup is weakly normal-homomorph-containing
  2. Weakly normal-homomorph-containing implies strictly characteristic

Proof

The proof follows directly by piecing together facts (1) and (2).