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 for any 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

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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
order-unique subgroup	finite subgroup and no other subgroup of the same order	(via isomorph-free)	(via isomorph-free)	Isomorph-free subgroup|FULL LIST, MORE INFO
isomorph-free subgroup	no other isomorphic subgroup	(obvious)	any non-co-Hopfian group, such as the group of integers, as a subgroup of itself	Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
isomorph-containing subgroup	contains every isomorphic subgroup			Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
homomorph-containing subgroup	contains every homomorphic image			Intermediately fully invariant subgroup, Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
subisomorph-containing subgroup	contains every subgroup isomorphic to a subgroup of it			Isomorph-containing subgroup, Transfer-closed characteristic subgroup|FULL LIST, MORE INFO
intermediately fully invariant subgroup	fully invariant in every intermediate subgroup			Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup	(obvious)		|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
intersection of finitely many intermediately characteristic subgroups	intersection of finitely many intermediately characteristic subgroups	(obvious)	follows from intermediate characteristicity is not finite-intersection-closed	|FULL LIST, MORE INFO
sub-intermediately characteristic subgroup	there is a chain of subgroups from the subgroup to the whole group, with each member intermediately characteristic in its successor	(obvious))	follows from intermediate characteristicity is not transitive	|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	(obvious)	characteristicity does not satisfy intermediate subgroup condition	Intersection of finitely many intermediately characteristic subgroups|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 $K$.
finite-intersection-closed subgroup property	No	intersection of two isomorph-free subgroups need not be intermediately characteristic		It is possible to have a group $G$ and intermediately characteristic subgroups $H, K$ of $G$ such that $H \cap K$ is not intermediately characteristic in $G$ (i.e., there exists $L \le G$ containing $H \cap K$ as a non-characteristic subgroup). In fact, we can choose an example where both $H$ and $K$ are isomorph-free in $G$.
strongly join-closed subgroup property	Yes	intermediate characteristicity is strongly join-closed		Suppose $H_i, i \in I$ is a family of intermediately characteristic subgroups of a group $G$. Then, thee join of subgroups $\langle H_i \rangle_{i \in I}$ is also intermediately characteristic in $G$.

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.