Combinatorial group theory
Encyclopedia
In mathematics
, combinatorial group theory is the theory of free group
s, and the concept of a presentation of a group
by generators and relations. It is much used in geometric topology
, the fundamental group
of a simplicial complex
having in a natural and geometric way such a presentation.
A very closely related topic is geometric group theory
, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.
It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups
; and the classical Burnside problem.
A proto-form is found in the 1856 Icosian Calculus
of William Rowan Hamilton
, where he studied the icosahedral symmetry
group via the edge graph of the dodecahedron.
The foundations of combinatorial group theory were laid by Walther von Dyck
, student of Felix Klein
, in the early 1880s, who gave the first systematic study of groups by generators and relations.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, combinatorial group theory is the theory of free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
s, and the concept of a presentation of a group
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
by generators and relations. It is much used in geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of a simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
having in a natural and geometric way such a presentation.
A very closely related topic is geometric group theory
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.
It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
; and the classical Burnside problem.
History
See for a detailed history of combinatorial group theory.A proto-form is found in the 1856 Icosian Calculus
Icosian Calculus
The Icosian Calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations....
of William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
, where he studied the icosahedral symmetry
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
group via the edge graph of the dodecahedron.
The foundations of combinatorial group theory were laid by Walther von Dyck
Walther von Dyck
Walther Franz Anton von Dyck , born Dyck and later ennobled, was a German mathematician...
, student of Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, in the early 1880s, who gave the first systematic study of groups by generators and relations.