Subgroup whose normalizer equals its normal closure

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: tightly 2-subnormal subgroup and 2-hypernormalized subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup $H$ of a group $G$ is termed a subgroup whose normalizer equals its normal closure if it satisfies the following equivalent conditions:

1. The normalizer $N_{G}(H)$ of $H$ in $G$ equals the normal closure $H^{G}$ of $H$ in $G$.
2. It is both a 2-hypernormalized subgroup (its normalizer is normal) and a subgroup with unique 2-subnormal series (it has a unique subnormal series of length two).

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
2-hypernormalized subgroup	normalizer is normal			|FULL LIST, MORE INFO
Subgroup with unique 2-subnormal series	has unique subnormal series of length two; equivalently, its normal closure is the normal core of its normalizer			|FULL LIST, MORE INFO
2-subnormal subgroup	normal subgroup of normal subgroup			|FULL LIST, MORE INFO