Subgroup whose focal subgroup equals its derived subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup H of a group G is termed a subgroup whose focal subgroup equals its commutator subgroup if we have the following condition. Let \operatorname{Foc}_G(H) denote the focal subgroup of H in G:

\operatorname{Foc}_G(H) = \langle xy^{-1} \mid x,y \in H, \exists g \in G, gxg^{-1} = y \rangle.

Then, we require that:

\operatorname{Foc}_G(H) = [H,H],

i.e., the focal subgroup of H equals its own commutator subgroup.

Relation with other properties

Stronger properties