Farjoun's question

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Statement

Let $H$ be a finite nilpotent homomorphism-closed subgroup of a group $G$ . Then, $H$ must be equal to $G$ .

A variant of this is the claim that any finite nilpotent nearly closed subgroup of a group must be the whole group.

On a question of Farjoun by Michael Aschbacher, Finite Groups 2003 published by Walter de Gruyter