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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.
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RANDOM SUBGROUP PROPERTY: Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

This is an opposite of normality

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Weaker properties 2.2 Incomparable properties 2.3 Opposite properties 3 Importance 4 Examples 5 Metaproperties 5.1 Transitivity 5.2 Trimness 5.3 Intermediate subgroup condition

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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 H of a group G is termed conjugate-dense in G 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 G on the coset space G / H, 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

• Finite-dominating subgroup

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 M is a set with some additional structure, and G is the group of automorphisms with that additional structure. This G could in principle be very huge, and unmanageable. Now suppose adding some further structure to M causes the automorphism group to reduce to a much smaller subgroup H of G.

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

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.
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If are subgroups such that K is the union of conjugates of H within K, and G is the union of conjugates of K within G, then:

Every conjugate of K within G is expressible as a union of conjugates of H within G.

This forces H to be conjugate-dense in G. 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.