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From Groupprops

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Formalisms 2.1 In terms of the in-normalizer operator 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 4 Metaproperties 4.1 Intermediate subgroup condition

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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
VIEW RELATED: Subgroup property implications | | | | |
RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

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 all 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

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition