Subgroup defined up to isoclinism

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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 of a group is termed a subgroup defined up to isoclinism if it satisfies the following condition: It either contains the center or is contained in the derived subgroup (or both).

The significance of this is that any isoclinism between groups establishes a bijection between the subgroups defined up to isoclinism of one group and the subgroups defined up to isoclinism of the other group.

Examples

  • In an abelian group, the only subgroups defined up to isoclinism are the trivial subgroup and the whole group.
  • In a centerless group, every subgroup is a subgroup defined up to isoclinism.
  • In a perfect group, every subgroup is a subgroup defined up to isoclinism.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
autoclinism-invariant subgroup
endoclinism-invariant subgroup