Burnside's theorem

Burnside's theorem
Copyrighted free use · source
Name	William Burnside
Birth date	1852
Death date	1927
Field	Mathematics
Known for	Group theory

Burnside's theorem

Burnside's theorem is a classical result in finite group theory that gives strong constraints on the structure of finite groups whose order has only two prime factors. The theorem asserts that any finite group of order p^a q^b for primes p and q is solvable, a property that places such groups within the chain of subgroups terminating in the trivial group. This theorem played a pivotal role in the development of the classification of finite simple groups and influenced later work by mathematicians associated with institutions such as University of Cambridge and Trinity College, Cambridge.

Statement of the theorem

Burnside's theorem states that any finite group of order p^a q^b, where p and q are primes and a, b are nonnegative integers, is a solvable group. The conclusion of solvability means the group admits a subnormal series whose successive quotients are cyclic of prime order, linking to concepts studied by Évariste Galois and formalized in the work of Camille Jordan and Otto Hölder. Burnside's criterion contrasts with properties of alternating groups such as A5, which is a nonabelian simple group of order 60 not of the form p^a q^b.

Historical context and motivation

The theorem was proved by William Burnside in the early 20th century, during a period when research at centers like University of Göttingen, École Normale Supérieure, and University of Cambridge advanced group theory. Burnside's work built on the legacy of Augustin-Louis Cauchy, Arthur Cayley, and Camille Jordan and responded to questions about possible orders of nonabelian simple groups raised by researchers at institutions such as Royal Society and communicated in journals like Proceedings of the London Mathematical Society. The theorem informed later programs culminating in the classification of finite simple groups, where contributors from institutes like Institute for Advanced Study and University of Chicago—including John G. Thompson and Walter Feit—used solvability results as foundational lemmas.

Proofs and variants

Burnside's original proof employed character theory developed by Frobenius and methods from the representation theory cultivated at places like University of Leipzig and Humboldt University of Berlin. Subsequent proofs used group actions and transfer techniques linked to work by Issai Schur, Frobenius, and William Rowan Hamilton's algebraic frameworks. Variants include the p^a q^b theorem formulations and results using the Feit–Thompson theorem machinery, though the Feit–Thompson theorem addresses odd order groups and was proved by Walter Feit and John G. Thompson. Modern expositions employ modules over C and characters as developed in texts from publishers associated with Cambridge University Press and lecture series at Massachusetts Institute of Technology.

Applications and consequences

Burnside's theorem restricts the possible structure of finite groups arising in algebraic contexts such as symmetry groups of polyhedral objects studied since Leonhard Euler and Johannes Kepler. It yields immediate corollaries about impossibility of simple nonabelian groups of order p^a q^b, influencing classification efforts by researchers affiliated with Princeton University and University of Oxford. The theorem also informs representations encountered in the theory of Galois groups over number fields explored by Évariste Galois and later by Emil Artin, and it has implications for permutation groups relevant to work by Helmut Wielandt and Camille Jordan.

Examples and counterexamples

Examples of solvable groups covered by the theorem include groups of order 6 (isomorphic to S_3), order 12 (such as the dihedral group D6), and order 20 (including semidirect products studied by Walther von Dyck). A key counterexample to a naive converse is the alternating group A5 of order 60 = 2^2·3·5, which is simple and therefore not solvable; its order involves three distinct primes and so lies outside the p^a q^b hypothesis. Historical examples considered by Burnside and contemporaries included groups constructed by Camille Jordan and examined in correspondence with mathematicians at Royal Society meetings.

Generalizations and related results

Generalizations connect Burnside's theorem to the Feit–Thompson theorem, which shows every finite group of odd order is solvable, and to results on groups with restricted prime spectra studied by László Kovács and Michael Aschbacher. The theorem also relates to Sylow theory introduced by Ludwig Sylow and to transfer and cohomology techniques developed by Claude Chevalley and Jean-Pierre Serre. Contemporary research explores analogous restrictions for linear groups over finite fields considered by authors from Institute for Advanced Study and École Polytechnique and connects to the broader structural classification initiated at institutions like University of Cambridge and Harvard University.

Category:Theorems in group theory