Intersection of finitely many intermediately characteristic subgroups: Difference between revisions

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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''intersection of finitely many intermediately characteristic subgroups''' in <math>G</math> if there exists a collection <math>H_1, H_2, \dots, H_n</math> of <math>G</math> such that <math>H = \bigcap_{i=1}^n H_i</math>, and such that each <math>H_i</math> is an [[defining ingredient::intermediately characteristic subgroup]] of <math>G</math>.
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''intersection of finitely many intermediately characteristic subgroups''' in <math>G</math> if there exists a collection <math>H_1, H_2, \dots, H_n</math> of <math>G</math> such that <math>H = \bigcap_{i=1}^n H_i</math>, and such that each <math>H_i</math> is an [[defining ingredient::intermediately characteristic subgroup]] of <math>G</math>.
==Formalisms==
{{obtainedbyapplyingthe|finite intersection operator|intermediately characteristic subgroup}}


==Relation with other properties==
==Relation with other properties==

Latest revision as of 23:42, 31 May 2020

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

A subgroup H of a group G is termed an intersection of finitely many intermediately characteristic subgroups in G if there exists a collection H1,H2,…,Hn of G such that H=⋂i=1nHi, and such that each Hi is an intermediately characteristic subgroup of G.

Formalisms

In terms of the finite intersection operator

This property is obtained by applying the finite intersection operator to the property: intermediately characteristic subgroup
View other properties obtained by applying the finite intersection operator

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
intermediately characteristic subgroup characteristic in every intermediate subgroup (obvious) intermediate characteristicity is not finite-intersection-closed |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup invariant under all automorphisms follows from characteristicity is strongly intersection-closed |FULL LIST, MORE INFO
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