Quotient-powering-invariant characteristic subgroup

From Groupprops
Jump to: navigation, search
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

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