subnormal subgroup

subnormal subgroup
Name	Subnormal subgroup
Field	Group theory
Introduced by	Otto Hölder
Related concepts	Normal subgroup, Solvable group, Nilpotent group

subnormal subgroup A subnormal subgroup is a subgroup that can be reached from a group by a finite chain of intermediate subgroups each normal in the next. In group theory and the theory of finite groups, subnormality generalizes normal subgroup and interacts with central series, commutator series, and structural results about simple groups and solvable groups. The concept appears in classical work by Otto Hölder, Issai Schur, and is used in modern classifications such as the Feit–Thompson theorem and the Classification of finite simple groups.

Definition

A subgroup H of a group G is called subnormal in G if there exists a finite chain H = H_0 ⊲ H_1 ⊲ ... ⊲ H_n = G where each H_i is a normal subgroup of H_{i+1}. The minimal possible n in such a chain is the subnormal length (or defect) of H in G. This notion was studied in contexts including work of Emmy Noether on rings with group actions and in structural investigations by John G. Thompson and Bertram Huppert.

Basic Properties and Examples

Every normal subgroup is subnormal of length 1; every subgroup of a nilpotent group is subnormal because of the central series structure used in proofs by Philip Hall. In finite p-groups and metabelian groups arising in the work of Marshall Hall Jr. and Gustav A. Hedlund, subnormality often coincides with other subgroup properties. For classical matrix groups like GL(n, F), many algebraic subgroups are not subnormal; by contrast, Sylow subgroups in finite groups studied by Sylow and in contexts of Frobenius groups frequently provide natural subnormal examples. Subnormality is transitive: if K is subnormal in H and H is subnormal in G then K is subnormal in G, a fact used in proofs by Isaacs, I. M. and Gorenstein, Daniel.

Subnormal Series and Subnormal Length

A subnormal series is the finite chain of subgroups witnessing subnormality. The subnormal length n provides a numerical invariant analogous to derived length in solvable groups or nilpotency class in nilpotent groups. Results by Zassenhaus and Wielandt relate subnormal length to composition series and chief series used in the Jordan–Hölder theorem. In finite group theory, bounding subnormal length can be critical in arguments like those in Feit–Thompson theorem proofs and in the Odd Order Theorem literature.

Relationship with Normality and Other Subgroup Properties

Subnormality refines normality: normal implies subnormal, but the converse fails in general; classical counterexamples appear in groups investigated by Wielandt and in permutation group studies by Cayley and Jordan, Camille. Subnormality interacts with pronormality, abnormality, and self-normalizing subgroup concepts that appear in the works of Hall, Marshall and Frobenius. In simple groups, nontrivial proper subnormal subgroups cannot exist; this fact is central to the classification of finite simple groups and is used in arguments by Burnside, William and Thompson, John G..

Behavior under Group Operations (Products, Intersections, Quotients, Extensions)

Intersections of subnormal subgroups need not be subnormal in arbitrary groups; counterexamples are constructed in permutation groups related to work by Wielandt and Frobenius. Finite products and joins of subnormal subgroups can fail to be subnormal; however, in solvable groups and nilpotent groups closure properties are stronger, as explored by Hall, Philip and Baer, Reinhold. For quotients: images of subnormal subgroups under group homomorphisms are subnormal in the image, a fact applied in many structural reductions by Gorenstein, Daniel and Aschbacher, Michael. In extension problems considered in the literature of Schur, Issai and Hopf, Heinz, subnormality behaves well under central extensions but can be subtle for non-split extensions examined in cohomological treatments by Eilenberg, Samuel and Mac Lane, Saunders.

Important Theorems and Results

Wielandt’s theorems give criteria for subnormality in permutation groups and link subnormality to primitive action results celebrated in the work of Wielandt and Jordan, Camille. The Lenski–Robinson results and related work by Baer give characterizations of subnormality in certain linear groups. The Kegel–Wielandt theorem and results by Kegel, O. H. concern products of nilpotent subnormal subgroups and have been applied in proofs by Kourovka compilers. Theorems relating subnormal length to derived length appear in papers by Zassenhaus and Huppert, Bertram. Applications of subnormal subgroup theory appear across finite group classification, representation theory as in Brauer, Richard's work, and in modern treatments by authors such as Aschbacher, Michael, Guralnick, Robert and Robinson, Geoffrey.

Category:Group theory