Normal subgroup whose focal subgroup equals its derived subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup whose focal subgroup equals its commutator subgroup
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Definition

A normal subgroup whose focal subgroup equals its commutator subgroup is a subgroup H of a group G satisfying the following equivalent conditions:

  1. H is a normal subgroup of G and [H,H] = \operatorname{Foc}_G(H), i.e., H is a subgroup whose focal subgroup equals its commutator subgroup.
  2. [G,H] = [H,H].

Relation with other properties

Stronger properties

Weaker properties