Intermediately normal-to-characteristic subgroup

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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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Definition

Symbol-free definition

A subgroup of a group is termed intermediately normal-to-characteristic if it satisfies the following equivalent conditions:

• Whenever it is normal in any intermediate subgroup, it is also characteristic in the intermediate subgroup.
• It is an intermediately characteristic subgroup in its normalizer.

Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately normal-to-characteristic in $G$ if it satisfies the following equivalent conditions:

• For any subgroup $K$ of $G$ containing $H$ such that $H$ is normal in $K$, $H$ is characteristic in $K$.
• $H$ is characteristic in any subgroup of $G$ contained in its normalizer $N_{G}(H)$.

Formalisms

In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: intermediately characteristic subgroup
View other properties obtained by applying the in-normalizer operator

A subgroup $H\leq G$ is intermediately normal-to-characteristic in $G$ if and only if $H$ is an intermediately characteristic subgroup in $N_{G}(H)$.

Relation with other properties

Stronger properties

• Intermediately automorph-conjugate subgroup
• Sylow subgroup
• Hall subgroup
• Join of intermediately automorph-conjugate subgroups

Weaker properties

• Intermediately subnormal-to-normal subgroup

Metaproperties

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