Conjugacy-closed subgroup

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Origin

The notion of conjugacy-closed subgroup was introduced in a journal article in the 1950s.

Definition

Symbol-free definition

A subgroup of a group is termed conjugacy-closed if any two elements of the subgroup that are conjugate in the whole group are also conjugate in the subgroup. A conjugacy-closed subgroup is also termed c-closed.

Definition with symbols

A subgroup $H$ of a group $G$ is termed conjugacy-closed if given $x$ and $y$ in $H$ such that there is $g$ in $G$ satisfying $gx{g}^{-1}=y$, then there is an $h$ in $H$ satisfying $hx{h}^{-1}=y$.

In terms of restriction formalisms

The property of being conjugacy-closed arises via the relation restriction formalism with both the left and right properties being the equivalence relation of being conjugate.

Relation with other properties

Conjunction with other properties

• Conjugacy-closed normal subgroup

Stronger properties

• Direct factor
• Central factor

Facts

General linear groups of subfields are conjugacy-closed

This is a typical context in which conjugacy-closedness comes up in representation theory. Let $k$ be a subfield of $K$. Then, we get a natural embedding of $G{l}_n(k)$ in $G{L}_n(K)$.

It turns out that for this embedding, $G{L}_n(k)$ is a conjugacy-closed subgroup of $G{L}_n(K)$. The proof relies on basic facts in linear algebra, and is often used implicitly or explicitly in proofs, when we simply talk of two linear transformations being "conjugate" without specifying whether we are thinking of them as conjugate in the smaller field or in the bigger 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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Any conjugacy-closed subgroup of a conjugacy-closed subgroup is conjugacy-closed. This follows from the fact that conjugacy-closedness is a balanced subgroup property with respect to the relation restriction formalism. For full proof, refer: Conjugacy-closedness is transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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• The property of being conjugacy-closed is trivially true, that is, the trivial subgroup is always conjugacy-closed.
• The property of being conjugacy-closed is identity-true, that is, the whole group is conjugacy-closed as a subgroup of itself.

Intersection-closedness

It is not clear whether an intersection of conjugacy-closed subgroups is conjugacy-closed.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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The property of being conjugacy-closed satisfies the intermediate subgroup condition. This is because the equivalence relation of being conjugate in a smaller subgroup implies the equivalence relation of being conjugate in the whole group. For full proof, refer: Conjugacy-closedness satisfies intermediate subgroup condition