Intermediately local powering-invariant subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

A subgroup H of a group G is termed intermediately local powering-invariant in G if, for every intermediate subgroup K of G (i.e., H \le K \le G), H is a local powering-invariant subgroup of K.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite subgroup |FULL LIST, MORE INFO
periodic subgroup |FULL LIST, MORE INFO
local divisibility-closed subgroup |FULL LIST, MORE INFO
retract (via local divisibility-closed) (via local divisibility-closed) |FULL LIST, MORE INFO
direct factor (via retract) (via retract) |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
intermediately powering-invariant subgroup powering-invariant in every intermediate subgroup. |FULL LIST, MORE INFO
local powering-invariant subgroup |FULL LIST, MORE INFO
powering-invariant subgroup Intermediately powering-invariant subgroup|FULL LIST, MORE INFO

Facts

Formalisms

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: local powering-invariant subgroup
View other properties obtained by applying the intermediately operator