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]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity
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 of a group
is said to be intermediately characteristic if for any intermediate subgroup
(such that
),
is characteristic in
.
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
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
Metaproperties
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
quotient-transitive subgroup property | Yes | intermediate characteristicity is quotient-transitive | Suppose we have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
intermediate subgroup condition | Yes | (obvious) | 0 | Suppose ![]() ![]() ![]() ![]() ![]() |
finite-intersection-closed subgroup property | No | intersection of two isomorph-free subgroups need not be intermediately characteristic | It is possible to have a group ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
strongly join-closed subgroup property | Yes | intermediate characteristicity is strongly join-closed | Suppose ![]() ![]() ![]() ![]() |
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.