Local powering-invariant normal subgroup of nilpotent group

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This article describes a property that arises as the conjunction of a subgroup property: local powering-invariant normal subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

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

  1. G is a nilpotent group and H is a local powering-invariant normal subgroup of G, i.e., H is both a local powering-invariant subgroup and a normal subgroup of G.
  2. G is a nilpotent group and H is a quotient-local powering-invariant subgroup of G.
  3. G is a nilpotent group and H is a quotient-torsion-freeness-closed subgroup of G.

Equivalence of definitions

Relation with other properties

Weaker properties

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