Conjugacy-closed subgroup: Difference between revisions

Revision as of 03:19, 3 March 2007

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]
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This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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

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.

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

@@ Line 8: / Line 8: @@
 ==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'''.
+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 <math>H</math> of a group <math>G</math> is termed ''conjugacy-closed'' if given <math>x</math> and <math>y</math> in <math>H</math> such that there is <math>g</math> in <math>G</math> satisfying <math>gxg^{-1} = y</math>, then there is an <math>h</math> in <math>H</math> satisfying <math>hxh^{-1} = y</math>.
+A subgroup <math>H</math> of a group <math>G</math> is termed '''conjugacy-closed''' if given <math>x</math> and <math>y</math> in <math>H</math> such that there is <math>g</math> in <math>G</math> satisfying <math>gxg^{-1} = y</math>, then there is an <math>h</math> in <math>H</math> satisfying <math>hxh^{-1} = y</math>.
 ===In terms of restriction formalisms===
@@ Line 24: / Line 25: @@
 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.
-{{proofat}[[Conjugacy-closedness is transitive]]}}
+{{proofat|[[Conjugacy-closedness is transitive]]}}
 {{trim}}