Subnormal subgroup

A subnormal subgroup weakens normality by allowing a finite sequence of normal extensions up to the ambient group. The relation retains enough transitivity and closure to describe nilpotent groups, Fitting subgroups, components, and minimal normal subgroups. Wielandt's theorem that the join of two subnormal subgroups of a finite group is again subnormal is the central closure result. ^{[@isaacs2008]}

Definition

The notation $H\triangleleft\triangleleft G$ is common. Each $$H_i$$ need only be normal in the next member $H_{i+1}$ , not in all of $$G$$ .

Elementary properties

Subnormality is transitive: chains for $H\triangleleft\triangleleft K$ and $K\triangleleft\triangleleft G$ concatenate. It is also preserved by homomorphic images and inverse images.

Wielandt's join theorem

This is the most important and least obvious closure property. The product of normal subgroups is visibly normal, but two subnormal subgroups need not normalize one another, so their series cannot simply be spliced.

Architecture of the proof

Induct on $$|G|$$ together with the two defects. If $$S$$ and $$T$$ lie in a common proper subgroup $$M<G$$ , the intersection theorem makes them subnormal in $$M$$ , and induction gives $\langle S,T\rangle\triangleleft\triangleleft M\leq G$ . The difficult case is therefore $\langle S,T\rangle=G$ with no common proper overgroup.

Now examine the penultimate members $$S_1,T_1$$ of subnormal series. If one is proper, induction can be applied in $$S_1T_1$$ or an appropriate intersection; if both equal $$G$$ , then $S,T\triangleleft G$ and the conclusion is immediate. Wielandt's zipper lemma rules out the remaining possibility that a subgroup is subnormal in every proper overgroup but not in $$G$$ : such a subgroup would have a unique maximal overgroup, and the conjugates arising in the induction contradict uniqueness. Finiteness supplies maximal overgroups and terminates the induction. ^{[@isaacs2008]}

Characterization of finite nilpotent groups

Subnormal pi-subgroups and local cores

This is the encyclopedia-level conclusion extracted from Isaacs, Problem 2A.1: a seemingly weak subnormality condition becomes containment in the largest global normal local subgroup.

Relation with the Fitting subgroup

Normal closures and minimal normal subgroups

If $S\triangleleft\triangleleft G$ is nonabelian simple, distinct conjugates of $$S$$ are either equal or commute elementwise. Its normal closure

S^G=\langle S^g\mid g\in G\rangle

is therefore a direct product of mutually commuting simple conjugates and is a minimal normal subgroup of $$G$$ . This exercise-derived result is the prototype for the later theory of components and the layer.

Permutable subgroups and the zipper lemma

Suppose $$S$$ permutes with each conjugate:

SS^x=S^xS\qquad(x\in G).

Then $$S$$ is subnormal in a finite group. A direct series is not visible from the set products. The proof uses Wielandt's zipper lemma: if $$S$$ is subnormal in every proper overgroup containing it but is not subnormal in $$G$$ , then $$S$$ has a unique maximal overgroup. Conjugate-permutability forces a contradiction to that uniqueness.

This article integrates the intersection theorem, Wielandt's join theorem, the Fitting criterion, and the structural conclusions of Problems 2A.1 and 2A.7--2A.8 from Isaacs. Historical attribution and terminology were cross-checked against Wielandt's paper and the cited public encyclopedia revision. ^{[@isaacs2008]} ^{[@wielandt1939]} ^{[@wikipedia-subnormal]}