Subgroup whose normalizer equals its normal closure

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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