Conjugate-dense subgroup

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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, 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 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

Opposite properties

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.