Conjugate-dense subgroup

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This is an opposite of normality

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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

Definition with symbols

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

• ${\bigcup }_{g\in G}gH{g}^{-1}=G$
• For any $a\in G$, there exists $b\in G$ such that $ba{b}^{-1}\in H$

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

Examples

• 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).

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 $H\le K\le G$ 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 Template:Identity-true.

Intermediate subgroup condition

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