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]

Definition

Symbol-free definition

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

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 \le 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

Weaker properties

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