# Conjugate-dense subgroup

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]

*This is an opposite of normality*

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article is about a general term. A list of important particular cases (instances) is available at Category:Instances of conjugate-dense subgroups

## Definition

### Symbol-free definition

A subgroup of a group is said to be **conjugate-dense** if it satisfies the following equivalent conditions:

- The union of all conjugates of the subgroup in the group, is the whole group
- Every element in the whole group is conjugate to some element in the subgroup
- Every cyclic subgroup of the whole group is conjugate to a cyclic subgroup of the given subgroup. In other words, it
*dominates*all cyclic subgroups - For the action of the whole group on the coset space, every element of the group has a fixed point.

### Definition with symbols

A subgroup of a group is termed **conjugate-dense** in if it satisfies the following equivalent conditions:

- For any , there exists such that
- For every cyclic subgroup , there exists such that
- Under the natural action of on the coset space , every has a fixed point.

## Relation with other properties

### Weaker properties

- Contranormal subgroup: Here, we only require that the normal closure should be the whole group. Note that the normal closure may, in general be much bigger than the union of conjugates.

### Incomparable properties

### Opposite properties

- Subgroup of finite index: It turns out that there cannot be a conjugate-dense subgroup of finite index other than the whole group.
`For full proof, refer: Union of all conjugates is proper` - Normal subgroup: Clearly, the only normal conjugate-dense subgroup is the whole group

## Importance

The general context in which being conjugate-dense is important is as follows. Suppose is a set with some additional structure, and is the group of automorphisms with that additional structure. This could in principle be very huge, and unmanageable. Now suppose adding some *further* structure to causes the automorphism group to reduce to a much smaller subgroup of .

In principle we could lose a lot of the symmetry in when we pass to . Thus, we are often interested in the question: when can we guarantee that every is conjugate (in ) to some element of ? In other words, is conjugate-dense in ? If the answer to this question is *yes*, then that means that at least if we are looking at only one element of at a time, then we might safely assume that our element is in .

Examples of passage to additional structure are:

- Giving a differential structure to a topological manifold
- Giving a Riemannian structure to a topological or differential manifold (typically, the isometry group of a Riemannian manifold is far from conjugate-dense in the group of diffeomorphisms or homeomorphisms)
- Giving a linear or piecewise linear structure (or a simplicial structure) to a topological manifold

The importance of finite-dominating subgroups is for similar reasons.

## Examples

For a full list of examples, refer:

Category:Instances of conjugate-dense subgroups

- The subgroup of upper triangular matrices, viz the Borel subgroup , is conjugate-dense in : in other words, every invertible matrix is conjugate to an upper triangular matrix (the analogous result does not hold for a non-algebraically closed field, and in particular, it doesn't hold for any finite field).
`Further information: Triangulability theorem` - The statement that every element of is a rotation about some axis (called Euler's theorem) can be rephrased as saying that is conjugate-dense in
`Further information: Euler's theorem`

## 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

If are subgroups such that is the union of conjugates of within , and is the union of conjugates of within , then:

Every conjugate of within is expressible as a union of conjugates of within .

This forces to be conjugate-dense in . `For full proof, refer: Conjugate-denseness is transitive`

### Trimness

The property of being conjugate-dense can be satisfied by the trivial subgroup only if the whole group is trivial. However, it is always satisfied by the whole group, hence it is identity-true subgroup property.

### Intermediate subgroup condition

The property of being conjugate-dense probably does not satisfy the intermediate subgroup condition.