Quotient-powering-invariant characteristic subgroup
From Groupprops
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: quotient-powering-invariant subgroup and characteristic subgroup
View other subgroup property conjunctions | view all subgroup properties
Contents
Definition
A subgroup of a group is termed a quotient-powering-invariant characteristic subgroup if it is both a quotient-powering-invariant subgroup and a characteristic subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| characteristic subgroup of abelian group | characteristic subgroup of abelian group is quotient-powering-invariant | Characteristic subgroup of center|FULL LIST, MORE INFO | ||
| complemented characteristic subgroup | follows from complemented normal implies quotient-powering-invariant | |FULL LIST, MORE INFO | ||
| finite characteristic subgroup | finite normal implies quotient-powering-invariant and characteristic implies normal | |FULL LIST, MORE INFO | ||
| characteristic subgroup of finite index | normal of finite index implies quotient-powering-invariant and characteristic implies normal | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| powering-invariant characteristic subgroup | |FULL LIST, MORE INFO | |||
| quotient-powering-invariant subgroup | |FULL LIST, MORE INFO | |||
| powering-invariant normal subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||
| powering-invariant subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||
| characteristic subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO | |||
| normal subgroup | Powering-invariant characteristic subgroup|FULL LIST, MORE INFO |
