Conjugacy-closed subgroup

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

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 sometimes termed c-closed, but now c-closed subgroup has a different meaning.

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 $$gxg^{-1} = y$$, then there is an $$h$$ in $$H$$ satisfying $$hxh^{-1} = y$$.

A related term is fusion. Two elements of a subgroup are said to fuse in the whole group if they become conjugate in the whole group. A subgroup is conjugacy-closed, essentially if no further fusion occurs when passing from the subgroup to the whole group.

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.

Conjunction with other properties

 * Weaker than::Conjugacy-closed normal subgroup
 * Weaker than::Sylow retract: This is the conjunction with the property of being a Sylow subgroup. It turns out that a conjugacy-closed Sylow subgroup must be a retract (i.e., possesses a normal complement).

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Central factor
 * Weaker than::Retract: However, for a Sylow subgroup, the property of being a retract and being a conjugacy-closed subgroup are equivalent.

Incomparable properties

 * Conjugate-dense subgroup: A subgroup is conjugate-dense if any element of the group is conjugate to some element of the subgroup. There is a close relation between conjugacy-closed subgroups and conjugate-dense subgroups, even though neither property implies the other.

Examples
For a full list of examples, refer:

Category:Instances of conjugacy-closed subgroups

For a full list of non-examples, refer:

Category:Instances of non-conjugacy-closed subgroups

General linear groups over 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 $$Gl_n(k)$$ in $$GL_n(K)$$.

It turns out that for this embedding, $$GL_n(k)$$ is a conjugacy-closed subgroup of $$GL_n(K)$$. 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.

Symmetric group is conjugacy-closed in the general linear group
This is again a deep and important fact, true only when the field has characteristic zero.

Symmetric groups on subsets are conjugacy-closedness
Suppose $$A$$ is a subset of $$B$$. Then, the symmetric group of $$A$$ embeds inside the symmetric group on $$B$$, and under this embedding, it is a conjugacy-closed subgroup. This follows from the fact that cycle type determines conjugacy class for a permutation.

A general strategy to proving conjugacy-closedness
The following is a general strategy to showing that a subgroup $$H$$ is conjugacy-closed in a group $$G$$. We find a subgroup $$K$$ of $$H$$ such that:


 * $$K$$ is conjugate-dense in $$H$$
 * Any two elements of $$K$$ which are conjugate in $$G$$, are in fact conjugate in $$H$$

More generally, $$K$$ need not even be a subgroup; we need to find a subset of $$H$$ such that the union of its conjugates is $$H$$, and such that any two elements of the subset which are conjugate in $$G$$, are conjugate in $$H$$.

Metaproperties
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.


 * 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.

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.

If $$H \le G$$ and $$K_1, K_2$$ are intermediate subgroups of $$G$$ containing $$H$$, such that $$H$$ is conjugacy-closed in both $$K_1$$ and $$K_2$$, then $$H$$ need not be conjugacy-closed in the join $$\langle K_1, K_2 \rangle$$.