Conjugate-dense subgroup

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This is an opposite of normality

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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:

• $\bigcup_{g \in G} gHg^{-1} = G$
• 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.

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 $g \in G$ 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). Further information: Triangulability theorem
• 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 $SO(3,\R)$ 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 transitive
ABOUT 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 $H \le K \le G$ 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 $G$. 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.