Divisibility-closed subgroup of nilpotent group: Difference between revisions

Latest revision as of 21:06, 1 April 2013

This article describes a property that arises as the conjunction of a subgroup property: divisibility-closed 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 divisibility-closed subgroup of nilpotent group if $G$ is a nilpotent group and $H$ is a divisibility-closed subgroup of $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
completely divisibility-closed subgroup of nilpotent group
divisibility-closed subgroup of abelian group

Weaker properties

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

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-| [[Weaker than::verbal subgroup of nilpotent group]] || || [[verbal subgroup of nilpotent group implies divisibility-closed]] (proof details pending) || || {{intermediate notions short|divisibility-closed subgroup of nilpotent group|verbal subgroup of nilpotent group}}
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