Conjugacy-closed normal subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: conjugacy-closed subgroup and normal subgroup
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Definition

Symbol-free definition

A subgroup of a group is termed conjugacy-closed normal if it satisfies the following equivalent conditions:

  • It is normal as well as conjugacy-closed (viz any two elements of the subgroup that are conjugate in the whole group are also conjugate in the subgroup).
  • Every inner automorphism of the whole group restricts to a class automorphism of the subgroup.

Definition with symbols

A subgroup H of a group G is termed conjugacy-closed normal if it satisfies the following equivalent conditions:

  • H \triangleleft G and whenever gxg^{-1} = y for x,y \in H and g \in G, there exists h \in H such that hxh^{-1} = y.
  • Given any automorphism \sigma of G such that for every x \in G, there exists g \in G such that \sigma(x) = gxg^{-1}, we also have the following: for every x \in H there exists h \in H such that \sigma(x) = hxh^{-1}.

Formalisms

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Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression H is a conjugacy-closed normal subgroup of G if ... This means that conjugacy-closed normality is ... Additional comments
class-preserving automorphism \to class-preserving automorphism every class-preserving automorphism of G restricts to a class-preserving automorphism of H the balanced subgroup property for class-preserving automorphisms Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism \to class-preserving automorphism every inner automorphism of G restricts to a class-preserving automorphism of H a left-inner subgroup property.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

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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 is conjugacy-closed normal in K and K is conjugacy-closed normal in G, then H is conjugacy-closed normal in G. This follows directly from conjugacy-closed normality being a balanced subgroup property in the function restriction formalism.

Intersection-closedness

It is not clear whether the intersection of two conjugacy-closed normal subgroups is conjugacy-closed normal.

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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Clearly, every group is conjugacy-closed normal as a subgroup of itself. Further, the trivial subgroup is also clearly conjugacy-closed normal in the whole group.

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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If H is conjugacy-closed normal in G, it is also conjugacy-closed normal in every intermediate subgroup K. This follows from these two facts:

  • If H is conjugacy-closed in G, H is conjugacy-closed in any intermediate subgroup K of G.
  • If H is normal in G, H is noraml in every intermediate subgroup K of G.