# Subgroup defined up to isoclinism

From Groupprops

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 |