Conjugate-dense subgroup

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:


 * $$\bigcup_{g \in G} gHg^{-1} = G$$
 * For any $$a \in G$$, there exists $$b \in G$$ such that $$bab^{-1} \in H$$
 * For every cyclic subgroup $$C \le G$$, there exists $$b \in G$$ such that $$bCb^{-1} \le H$$
 * Under the natural action of $$G$$ on the coset space $$G/H$$, every $$g \in G$$ has a fixed point.

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.
 * 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 $$g \in G$$ 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 $$B(n,\mathbb{C})$$, is conjugate-dense in $$GL(n,\mathbb{C})$$: 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).
 * The statement that every element of $$SO(3,\R)$$ is a rotation about some axis (called Euler's theorem) can be rephrased as saying that $$SO(2,\R)$$ is conjugate-dense in $$SO(3,\R)$$

Metaproperties
If $$H \le K \le G$$ 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 $$G$$.

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.

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