Subgroup defined up to isoclinism
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]
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.
- 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
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|