Abelian fully invariant subgroup

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This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 fully invariant subgroup or fully invariant abelian subgroup if H is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of G) and is also a fully invariant subgroup (or fully characteristic subgroup) of G.

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian characteristic subgroup abelian and a characteristic subgroup -- invariant under all automorphisms follows from fully invariant implies characteristic follows from characteristic not implies fully invariant in finite abelian group |FULL LIST, MORE INFO
abelian normal subgroup abelian and a normal subgroup -- invariant under all inner automorphisms (via abelian characteristic, follows from characteristic implies normal) follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property Abelian characteristic subgroup|FULL LIST, MORE INFO
abelian subnormal subgroup abelian and a subnormal subgroup Abelian characteristic subgroup|FULL LIST, MORE INFO