Powering-invariant characteristic subgroup

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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 Characteristic subgroup of center, Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
complemented characteristic subgroup (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
finite characteristic subgroup (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO
characteristic subgroup of finite index (via quotient-powering-invariant characteristic) Quotient-powering-invariant characteristic subgroup|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