# Conjugacy-closed subgroup

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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]

## Origin

The notion of conjugacy-closed subgroup was introduced in a journal article in the 1950s.

## Definition

### Symbol-free definition

A subgroup of a group is termed **conjugacy-closed** if any two elements of the subgroup that are conjugate in the whole group are also conjugate in the subgroup. A conjugacy-closed subgroup is also sometimes termed **c-closed**, but now c-closed subgroup has a different meaning.

### Definition with symbols

A subgroup of a group is termed **conjugacy-closed** if given and in such that there is in satisfying , then there is an in satisfying .

A related term is **fusion**. Two elements of a subgroup are said to fuse in the whole group if they become conjugate in the whole group. A subgroup is conjugacy-closed, essentially if no further fusion occurs when passing from the subgroup to the whole group.
`Further information: Category:Fusion theorems`

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

The property of being conjugacy-closed arises via the relation restriction formalism with both the left and right properties being the equivalence relation of being conjugate.

## Relation with other properties

### Conjunction with other properties

- Conjugacy-closed normal subgroup
- Sylow retract: This is the conjunction with the property of being a Sylow subgroup. It turns out that a conjugacy-closed Sylow subgroup must be a retract (i.e., possesses a normal complement).
`For full proof, refer: Conjugacy-closed and Sylow implies retract`

### Stronger properties

- Direct factor
- Central factor
- Retract:
*For proof of the implication, refer Retract implies conjugacy-closed and for proof of its strictness (i.e. the reverse implication being false) refer Conjugacy-closed not implies retract*. However, for a Sylow subgroup, the property of being a retract and being a conjugacy-closed subgroup are equivalent.`For full proof, refer: Conjugacy-closed and Sylow implies retract`

### Incomparable properties

- Conjugate-dense subgroup: A subgroup is conjugate-dense if any element of the group is conjugate to some element of the subgroup. There is a close relation between conjugacy-closed subgroups and conjugate-dense subgroups, even though neither property implies the other.

## Examples

For a full list of examples, refer:

Category:Instances of conjugacy-closed subgroups

For a full list of non-examples, refer:

Category:Instances of non-conjugacy-closed subgroups

### General linear groups over subfields are conjugacy-closed

`Further information: general linear group over subfield is conjugacy-closed`

This is a typical context in which conjugacy-closedness comes up in representation theory. Let be a subfield of . Then, we get a natural embedding of in .

It turns out that for this embedding, is a conjugacy-closed subgroup of . The proof relies on basic facts in linear algebra, and is often used implicitly or explicitly in proofs, when we simply talk of two linear transformations being "conjugate" without specifying whether we are thinking of them as conjugate in the smaller field or in the bigger field.

### Symmetric group is conjugacy-closed in the general linear group

`Further information: Brauer's permutation lemma`

This is again a deep and important fact, true only when the field has characteristic zero.

### Symmetric groups on subsets are conjugacy-closedness

`Further information: Symmetric group on subset is conjugacy-closed`
Suppose is a subset of . Then, the symmetric group of embeds inside the symmetric group on , and under this embedding, it is a conjugacy-closed subgroup. This follows from the fact that cycle type determines conjugacy class for a permutation.

### A general strategy to proving conjugacy-closedness

`Further information: Proving conjugacy-closedness`

The following is a general strategy to showing that a subgroup is conjugacy-closed in a group . We find a subgroup of such that:

- is conjugate-dense in
- Any two elements of which are conjugate in , are in fact conjugate in

More generally, need not even be a subgroup; we need to find a subset of such that the union of its conjugates is , and such that any two elements of the subset which are conjugate in , are conjugate in .

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Any conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed. This follows from the fact that conjugacy-closedness is a balanced subgroup property with respect to the relation restriction formalism.
`For full proof, refer: Conjugacy-closedness is transitive`

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

- The property of being conjugacy-closed is trivially true, that is, the trivial subgroup is always conjugacy-closed.
- The property of being conjugacy-closed is identity-true, that is, the whole group is conjugacy-closed as a subgroup of itself.

### Intersection-closedness

It is not clear whether an intersection of conjugacy-closed subgroups is conjugacy-closed.

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

The property of being conjugacy-closed satisfies the intermediate subgroup condition. This is because the equivalence relation of being conjugate in a smaller subgroup implies the equivalence relation of being conjugate in the whole group. `For full proof, refer: Conjugacy-closedness satisfies intermediate subgroup condition`

### Upper join-closedness

NO:This subgroup property isnotupper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

If and are intermediate subgroups of containing , such that is conjugacy-closed in both and , then need not be conjugacy-closed in the join . `For full proof, refer: Conjugacy-closedness is not upper join-closed`