Powering-invariant central 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 central subgroup
View other subgroup property conjunctions | view all subgroup properties
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

A subgroup H of a group G is termed a powering-invariant central subgroup if it satisfies the following equivalent conditions:

  1. H is both a powering-invariant subgroup of G and a central subgroup of G.
  2. H is both a quotient-powering-invariant subgroup of G and a central subgroup of G.

Equivalence of definitions

Further information: central implies normal satisfying the subgroup-to-quotient powering-invariance implication

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant subgroup of abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
quotient-powering-invariant subgroup |FULL LIST, MORE INFO
powering-invariant normal subgroup |FULL LIST, MORE INFO