derived subgroup

derived subgroup
Name	Derived subgroup
Other names	Commutator subgroup
Notation	[G,G], G'
Field	Mathematics
Subfield	Group theory, Abstract algebra
Introduced by	Camille Jordan, Otto Hölder

derived subgroup

The derived subgroup is the subgroup generated by all commutators in a group, central to Camille Jordan's and Otto Hölder's work on permutation groups and composition series. It measures non-abelianness and appears in the analysis of Galois groups, Sylow theorems applications, and the classification of finite simple groups such as the Feit–Thompson theorem and the Classification of finite simple groups. The construction is functorial in many contexts used by researchers at institutions like the Institute for Advanced Study and in texts by authors such as John G. Thompson and Walter Ledermann.

Definition and basic properties

For a group G the derived subgroup is defined as the subgroup generated by all commutators [x,y]=x^{-1}y^{-1}xy for x,y in G; this concept appears in the work of Niels Henrik Abel and in Jordan's treatises on substitution groups. It is the smallest normal subgroup N of G with abelian quotient G/N, a fact used in studies related to the Jordan–Hölder theorem and the structure of symmetric groups. Fundamental properties include normality, characteristicity (invariant under automorphisms such as those studied by Emmy Noether), and the fact that the quotient G/[G,G] is the largest abelian quotient, a notion appearing in Galois theory and in connections to Hilbert's Theorem 90 in cohomological contexts.

Examples and computations

For an abelian group A, the derived subgroup is trivial, a fact exploited in the proofs of results by Évariste Galois concerning solvable groups. For the symmetric group S_n with n ≥ 3 the derived subgroup is the alternating group A_n, a classical computation used in permutation group theory and in work by Arthur Cayley and Augustin-Louis Cauchy. For a dihedral group D_{2n} the derived subgroup is cyclic of order dividing n, appearing in studies of polygonal symmetry in works by Joseph Fourier and later group theorists. Matrix groups such as GL(n,ℝ) and SL(n,ℝ) illustrate computations where commutator subgroups relate to determinant maps and to special linear groups, topics treated by Élie Cartan and Hermann Weyl. In finite p-groups treated by Philip Hall the derived subgroup often features in central series calculations and examples from Burnside problem contexts.

Relationship with other subgroup series

The derived subgroup is the first step of the derived series G^{(0)}=G, G^{(1)}=[G,G], G^{(n+1)}=[G^{(n)},G^{(n)}], a filtration used alongside the lower central series central to Marshall Hall Jr.'s and Wilhelm Magnus's work. It contrasts with the lower central series Γ_1=G, Γ_{n+1}=[Γ_n,G] used in nilpotency investigations by Philip Hall and John Milnor. Connections to the upper central series appear in structural theorems by I. Schur and are exploited in the classification efforts by authors like Daniel Gorenstein. The behavior of derived subgroups under formation of direct and semidirect products is a recurring theme in literature from Kurt Reidemeister to modern texts by Derek Holt and C. R. Leedham-Green.

Derived length and solvability

Iterating the derived subgroup yields the derived length, used to define solvable groups, a concept central to Évariste Galois's original work on polynomial solvability and to the Feit–Thompson theorem proving odd-order solvability. Finite solvable groups have finite derived length, a property exploited in proofs by John Thompson and Michael Aschbacher and in algorithms for computing group structure implemented by projects at the GAP and Magma computational algebra systems. Derived length distinguishes classes such as metabelian groups (derived length two), which appear in the theory developed by Otto Schreier and in studies of group extensions by H. F. Blichfeldt.

Functoriality and behavior under homomorphisms

The derived-subgroup construction is functorial: a homomorphism f:G→H maps commutators in G to commutators in H, so f([G,G])≤[H,H]. This naturality underlies applications in representation theory contexts studied by Issai Schur and in cohomology theories influenced by Claude Chevalley and Jean-Pierre Serre. For surjective homomorphisms φ:G→Q the image φ([G,G]) equals [Q,Q], a fact used in quotient analyses in proofs by Otto Hölder and in transfer theorems by John G. Thompson. Preimages satisfy f^{-1}([H,H]) being a normal subgroup containing [G,G], a property utilized in extension classification problems addressed by I. Schur and Eilenberg–MacLane-style cohomology studies.

Generalizations and variants

Variants include the transfinite derived series used in infinite group theory as in work by J. H. C. Whitehead and studies of hyperabelian groups by Bernhard Neumann. The notion extends to profinite groups and pro-p groups in research by John P. Serre and Leonard Ribes, where closures of commutator subgroups are considered. Related constructions such as the verbal subgroup generated by a set of group words, the marginal subgroup in cohomological contexts treated by Samuel Eilenberg and Saunders Mac Lane, and commutator subgroups in Lie algebra analogues studied by Sophus Lie and Nathan Jacobson appear across algebraic research. These variants are instrumental in modern classification, cohomology, and computational projects at institutions including the Mathematical Sciences Research Institute and in software like GAP.

Category:Group theory