Normal subgroup

Normal subgroup
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Normal subgroup
Type	Algebraic structure
Field	Group theory
Introduced	Évariste Galois
Key contributors	Niels Henrik Abel, Arthur Cayley, Camille Jordan, Emmy Noether

Normal subgroup

A normal subgroup is a subgroup invariant under conjugation within a group; it is central to constructions like quotient groups and the fundamental isomorphism theorems. Introduced in the development of permutation theory and Galois theory, the concept is intertwined with work by Évariste Galois, Niels Henrik Abel, and Camille Jordan and plays a role across problems studied in Galois's investigations, applications in Sylow theory, and structural classification pursued by Émile Mathieu and William Burnside.

Definition

Let G be a group and H a subgroup of G. H is called a normal subgroup if for every g in G and h in H the element g h g^{-1} lies in H. The condition can be stated as gHg^{-1} = H for all g in G, making H invariant under the conjugation action of G on itself, a viewpoint used in studies by Arthur Cayley and in the orbit-stabilizer framework linked to Augustin-Louis Cauchy and Évariste Galois.

Properties and equivalent characterizations

Several equivalent formulations appear in algebraic literature influenced by Emmy Noether and Richard Dedekind: - Coset equality: left cosets gH coincide with right cosets Hg for all g in G, a perspective used in quotient constructions by Camille Jordan and formalized in texts influenced by Emmy Noether. - Kernel characterization: H is the kernel of some group homomorphism f: G → K; kernels arise in the first isomorphism theorem developed alongside work by Leopold Kronecker and later systematized in structural algebra frameworks by Emmy Noether and Emil Artin. - Normality via action: H is fixed under the conjugation action of G on the set of its subgroups, a language that connects to permutation group methods of Sophus Lie and permutation representation studies by Arthur Cayley. - Commutator relation: G′, the commutator subgroup introduced by Otto Hölder and studied by Wilhelm Magnus, is a normal subgroup; its normality underpins solvability investigations initiated by Évariste Galois.

Examples and non-examples

Classical examples drawn from historic group families: - Center Z(G) (studied by Camille Jordan) and commutator subgroup G′ are always normal, providing building blocks used in the classification efforts of William Burnside. - Subgroups of index 2 in finite groups are normal, a fact used in arguments by Arthur Cayley and in early group enumeration by Édouard Mathieu. - Normal subgroups appear in symmetric and alternating groups: A_n is normal in S_n for n ≥ 3, a relation central to permutation group theory explored by Évariste Galois and Camille Jordan. Non-examples and pitfalls: - Many subgroups of S_n are not normal, a theme in counterexamples used by Sophus Lie and in the development of simple group theory culminating in work by Robert Griess. - Stabilizer subgroups in transitive actions need not be normal; such examples were central in orbit-stabilizer analyses by Arthur Cayley and in permutation group counterexamples cataloged by Jordan.

Quotient groups and homomorphism theorems

If N is normal in G, the set of cosets G/N carries a group structure with multiplication (gN)(hN) = (gh)N, a construction essential to the first isomorphism theorem attributed to developments by Leopold Kronecker and formalized in structural algebra by Emmy Noether. The correspondence theorem relates subgroups of G/N to subgroups of G containing N, a principle used in classification projects by William Burnside and in module-theoretic analogues studied by Emil Artin. Quotient groups enable formation of exact sequences and factor groups appearing in homological contexts developed by Hermann Weyl and in group cohomology work by Jean-Pierre Serre.

Conjugacy, cores, and normalizer relations

Conjugacy classes partition G; normal subgroups are unions of whole conjugacy classes, a fact used in character theory advanced by Frobenius and Issai Schur. The core of a subgroup H, defined as the intersection of all conjugates ⋂_{g∈G} gHg^{-1}, is the largest normal subgroup contained in H and is used in permutation group analysis by Camille Jordan and in block theory initiated by Richard Brauer. The normalizer N_G(H) = { g ∈ G : gHg^{-1} = H } controls when H is normal; if N_G(H) = G then H is normal. These relations are central in transfer and fusion problems studied by Alperin and Bender and in local analysis of finite groups used throughout the classification of finite simple groups by researchers including Daniel Gorenstein and John Thompson.

Series and related concepts (simple, solvable, composition)

Normal subgroups organize groups via series and factor chains. A composition series is a chain 1 = N_0 ◁ N_1 ◁ ... ◁ N_k = G with simple factors N_{i+1}/N_i, a concept formalized in the Jordan–Hölder theorem proven by Camille Jordan and Otto Hölder. Solvable groups, central to Évariste Galois's criterion for solvability by radicals, are those with a normal series whose factors are abelian; this notion guided classification efforts by William Burnside and influenced later work by Emil Artin. Simple groups, having no nontrivial proper normal subgroups, lie at the core of classification projects culminating with contributions from Daniel Gorenstein, Robert Griess, and the broader finite simple group project.

Category:Group theory