Abelian hereditarily normal subgroup

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This article describes a property that arises as the conjunction of a subgroup property: hereditarily normal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions
This article describes a property that arises as the conjunction of a subgroup property: transitively normal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions

Definition

A subgroup H of a group G is termed an abelian hereditarily normal subgroup or abelian transitively normal subgroup of G if it satisfies the following equivalent conditions:

  1. H is abelian as a group and is a hereditarily normal subgroup of G.
  2. H is abelian as a group and is a transitively normal subgroup of G.
  3. H is an abelian normal subgroup of G and the induced action of the quotient group G/H on H is by power automorphisms.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Central subgroup
Cyclic normal subgroup

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hereditarily normal subgroup
Transitively normal subgroup