group theory
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| group theory | |
|---|---|
| Name | Group theory |
| Field | Mathematics |
| Subfields | Algebra, Abstract algebra |
| Notable concepts | Group, Subgroup, Homomorphism, Normal subgroup, Quotient group |
| Notable people | Évariste Galois, Niels Henrik Abel, Camille Jordan, Emmy Noether |
group theory is the mathematical study of algebraic structures called groups, which formalize the concept of symmetry and invertible operations. It provides foundational tools across Mathematics, informing work in Number theory, Topology, Geometry, and mathematical physics, and connects to computational problems in Computer science and crystallography in Materials science.
Definition and Basic Concepts
A group is a set with a binary operation satisfying closure, associativity, an identity element, and inverses; central accompanying notions include Subgroup, Cyclic group, Abelian group, and order. Core constructions are generated by sets (see Generating set), presented by relations (see group presentation), and distinguished by properties such as being normal or simple. Important invariants include center, commutator subgroup, and the concept of solvability (as in solvable groups).
Examples and Classification of Groups
Classic examples are finite symmetric groups like S_n, alternating groups like A_n, matrix groups such as GL(n, F), SL(n, F), O(n), and U(n), as well as infinite groups like Integers under addition and free groups. Classification results include the classification of finite simple groups with families like cyclic simple groups, groups of Lie type, and 26 sporadic groups including the Monster. Other families include p-groups and solvable families studied via Sylow theorems and Burnside's paqb theorem.
Group Homomorphisms and Quotients
Morphisms between groups are homomorphisms; kernels and images give rise to isomorphism theorems such as the First isomorphism theorem and Third isomorphism theorem. Quotients by normal subgroups produce quotient groups; extensions and exact sequences (e.g., Short exact sequence) encode how groups build from subgroups. Further structure is analyzed via automorphism groups, inner automorphisms, and cohomology concepts like Group cohomology for extension classification.
Group Actions and Permutation Groups
Groups act on sets via actions; stabilizers and orbits yield tools like the orbit–stabilizer theorem and Burnside's lemma for counting. Actions on combinatorial structures lead to permutation groups such as S_n and primitive groups studied in the O'Nan–Scott theorem. Geometric and topological actions connect to fundamental group actions on covering spaces and to transformation groups like Lie groups acting on manifolds, with fixed-point results such as the Lefschetz fixed-point theorem influencing dynamics.
Structure Theorems and Composition Series
Structural analysis uses series like composition series, Jordan–Hölder theorem, and chief series to decompose groups into simple factors. The Sylow theorems control p-subgroups, while the Feit–Thompson theorem and Burnside's theorem give constraints on solvability. For infinite groups, decomposition tools include HNN extensions and Free product with amalgamations; the theory of Tarski monsters and residually finite groups illustrates pathological and restrictive behaviors.
Representation Theory and Applications
Representation theory studies homomorphisms from groups into GL(V), enabling analysis via modules and characters, as in Character theory for finite groups and the Peter–Weyl theorem for compact groups. Linear representations connect to Quantum mechanics in Physics (symmetry of systems), to signal processing via Fourier transforms on groups, and to number-theoretic automorphic representations in the Langlands program. Computational applications occur in Cryptography and algorithmic group theory problems like the word problem.
Historical Development and Key Contributors
Origins trace to work by Évariste Galois on solvability of equations and by Niels Henrik Abel on insolvability, with further formalization by Camille Jordan and consolidation by William Rowan Hamilton and Arthur Cayley. Twentieth-century advances feature Emmy Noether's structural algebra, classification efforts culminating in the proof by contributors such as Daniel Gorenstein and John G. Thompson, and representation theory development by Frobenius and Issai Schur. Modern connections involve figures in Langlands research, and collaborative projects across institutions like Institute for Advanced Study and Mathematical Sciences Research Institute.