Intermediately 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 said to be intermediately characteristic if it is characteristic not only in the whole group but also in every intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be intermediately characteristic if forany intermediate subgroup $K$ (such that $H\le K\le G$), $H$ is characteristic in $K$.

Formalisms

In terms of the intermediately operator

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

The subgroup property of being intermediately characteristic can be obtained by applying the intermediately operator to the subgroup property of being characteristic.

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-free subgroup	no other isomorphic subgroup	(obvious)	any non-co-Hopfian group, such as the group of integers, as a subgroup of itself	|FULL LIST, MORE INFO
order-unique subgroup	finite subgroup and no other subgroup of the same order	(via isomorph-free)	(via isomorph-free)	|FULL LIST, MORE INFO
isomorph-containing subgroup	contains any isomorphic subgroup			|FULL LIST, MORE INFO
transfer-closed characteristic subgroup	intersection with any subgroup is characteristic in that subgroup	transfer-closed characteristic implies intermediately characteristic	intermediately characteristic not implies transfer-closed characteristic	|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	(obvious)	characteristicity does not satisfy intermediate subgroup condition	|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms (conjugations)	(via characteristic)	(via characteristic)	Characteristic subgroup|FULL LIST, MORE INFO

Related properties

• Image-closed characteristic subgroup

Metaproperties

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Here is a summary:

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
transitive subgroup property	No	intermediate characteristicity is not transitive		It is possible to have groups $H\le K\le G$ such that $H$ is intermediately characteristic in $K$ and $K$ is intermediately characteristic in $G$ but $H$ is not intermediately characteristic in $G$.
quotient-transitive subgroup property	Yes	intermediate characteristicity is quotient-transitive		Suppose we have groups $H\le K\le G$ such that $H$ is intermediately characteristic in $G$ and $K/H$ is intermediately characteristic in $G/H$. Then, $K$ is intermediately characteristic in $G$.
intermediate subgroup condition	Yes	(obvious)	0	Suppose $H\le K\le G$ are groups such that $H$ is intermediately characteristic in $G$. Then, $H$ is intermediately characteristic in $K$.

Effect of property operators

Right transiter

It turns out that any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is again intermediately characteristic. This follows from some simple reasoning and the fact that characteristicity is itself transitive. Further information: Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic

Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic.