Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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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.
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RANDOM SUBGROUP PROPERTY: Abelian normal subgroup: A subgroup that is Abelian as a group and normal as a subgroup.

Definition

A subgroup H of a group G is termed a commutator-in-centralizer subgroup if the commutator [G,H] is contained in the centralizer C_G(H), or equivalently, [[G,H],H] is trivial.,

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Commutator-in-center subgroup	commutator with whole group is contained in center of subgroup
Abelian normal subgroup				click here
Subgroup contained in centralizer of commutator subgroup	contained in the centralizer of the whole group's commutator subgroup
Aut-abelian normal subgroup				click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hereditarily 2-subnormal subgroup
Class two 2-subnormal subgroup
2-subnormal subgroup