Powering-invariant characteristic subgroup
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and characteristic subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a powering-invariant characteristic subgroup if it is both a powering-invariant subgroup and a characteristic subgroup.
Note that characteristic not implies powering-invariant. Also, a non-characteristic subgroup of a finite group is powering-invariant and not characteristic. Hence, neither property implies the other, so the conjunction is strictly stronger than both.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| quotient-powering-invariant characteristic subgroup | ||||
| characteristic subgroup of abelian group | characteristic subgroup of abelian group is powering-invariant | |FULL LIST, MORE INFO | ||
| complemented characteristic subgroup | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO | ||
| finite characteristic subgroup | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO | ||
| characteristic subgroup of finite index | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| powering-invariant normal subgroup | [powering-invariant subgroup]] as well as a normal subgroup | follows from characteristic implies normal | follows from finite examples for normal not implies characteristic | |FULL LIST, MORE INFO |
| powering-invariant subgroup | |FULL LIST, MORE INFO | |||
| characteristic subgroup | |FULL LIST, MORE INFO | |||
| normal subgroup | |FULL LIST, MORE INFO |