Subgroup whose focal subgroup equals its intersection with the 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

Suppose G is a group and H is a subgroup. Denote by \operatorname{Foc}_G(H) 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.

H is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup if we have:

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

Note that in general, we only have the containment \operatorname{Foc}_G(H) \le H \cap [G,G].

Relation with other properties

Stronger properties