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Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

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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.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions[SHOW MORE]

VIEW RELATED: Subgroup property implications |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property|
RANDOM SUBGROUP PROPERTY: Hall subgroup: A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.

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 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: |
ABOUT INTERMEDIATE SUBROUP CONDITION: View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition