Presentation of groups
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| Presentation of groups | |
|---|---|
| Name | Presentation of groups |
| Caption | Cayley graph illustrating a presentation by generators and relations |
| Field | Group theory |
| Introduced | 19th century |
| Notable figures | Augustin-Louis Cauchy, Niels Henrik Abel, Évariste Galois, Arthur Cayley, William Rowan Hamilton |
Presentation of groups.
A presentation of a group is a concise description by explicitly listing a set of generators and a set of relations among them, used throughout Group theory, Combinatorial group theory, Geometric group theory, Algebraic topology, and Low-dimensional topology. Presentations enable construction of groups such as Free group, Cyclic group, Dihedral group, and are central to algorithms developed by Max Dehn, John Conway, H. S. M. Coxeter, and researchers at institutions like Princeton University and University of Cambridge. Presentations connect to objects like the Cayley graph, Van Kampen theorem, Fundamental group, and computational systems including Todd–Coxeter algorithm, Knuth–Bendix completion algorithm, and implementations in software from Wolfram Research and SageMath.
Definition and Basic Concepts
A group presentation is denoted ⟨S | R⟩ where S is a set of generators and R is a set of relators, formal words in the generators; this formalism originates with work by Arthur Cayley and was systematized in Combinatorial group theory by Max Dehn and later by William Burnside. The free group on S, exemplified by Free group, maps onto the presented group with kernel the normal closure of R, a construction used in proofs by Emil Artin and Otto Schreier. Presentations relate to the Cayley graph construction and to algebraic invariants studied by I. M. Singer and E. H. Spanier, and to geometric realizations via the Universal cover and the Eilenberg–MacLane space K(G,1).
Presentations and Generators-Relations
Generators S may be finite or infinite; finite presentations appear in work of William Burnside and in classifications by H. S. M. Coxeter and John Milnor. Relations R can enforce torsion, commutativity, or more complex constraints as in presentations of Symmetric group, Alternating group, Heisenberg group, and Braid group. Standard moves on presentations include Tietze transformations introduced by Heinrich Tietze, Nielsen transformations related to Jakob Nielsen, and Andrews–Curtis moves connected to conjectures studied by James J. Andrews and Morton L. Curtis. Presentations underpin notions like deficiency and relator length used in research at Harvard University and University of Chicago.
Examples and Standard Presentations
Classical examples include ⟨a | a^n⟩ for the Cyclic group, ⟨r,s | r^n, s^2, srs = r^{-1}⟩ for the Dihedral group, ⟨x,y | x^2, y^3, (xy)^5⟩ for certain Triangle group instances studied in H. S. M. Coxeter’s work, and Artin presentations for Braid group investigated by Emil Artin. Presentations capture crystallographic groups classified by researchers associated with International Union of Crystallography and symmetry groups like Octahedral group and Icosahedral group. Infinite groups such as Baumslag–Solitar group, Lamplighter group, and the Grigorchuk group are often introduced via recursive or self-similar presentations in studies at Moscow State University and Steklov Institute.
Properties and Decision Problems
Algorithmic problems for presentations include the word problem, Conjugacy problem, and Isomorphism problem which were highlighted in work by Max Dehn and later by Emil Post and Andrey Markov. Undecidability results by Alonzo Church and Alan Turing and reductions by Pyotr Novikov and William Boone show general undecidability of the word problem for finitely presented groups; positive results occur for hyperbolic groups studied by Mikhail Gromov and for small cancellation groups developed by R. C. Lyndon and P. E. Schupp. Residual finiteness, growth rates studied by John Milnor and Grigori Perelman’s geometric techniques, and amenability considered by John von Neumann relate to properties detectable from specific classes of presentations.
Constructions and Operations on Presentations
Operations include free products with amalgamation and HNN extensions formalized by Sergei Adian and Hyman Bass, used to build groups with prescribed properties in work by H. S. M. Coxeter and Gilbert Baumslag. Presentations change under group extensions, direct products, semidirect products explored by Claude Chevalley, and via graph of groups decompositions examined by Jean-Pierre Serre in his study of Bass–Serre theory. Covering space techniques and applications of the Van Kampen theorem relate presentations to cell complexes and to 2-complexes studied by J. H. C. Whitehead.
Applications in Group Theory and Topology
Presentations serve to compute Fundamental groups of manifolds and knot complements following methods by Hermann Seifert and Walther van der Waerden and later refined by William Thurston. They are essential in constructing counterexamples in Geometric group theory and in proving rigidity theorems cited by Mostow rigidity and in the study of mapping class groups of Riemann surfaces developed by Harvey Cohn and Benson Farb. Computational group theory implementations in GAP and Magma use presentations for coset enumeration and homology calculations, with applications to classification problems pursued at École Normale Supérieure and Institute for Advanced Study.