In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, combinatorial group theory is the theory of
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...
s, and the concept of a
presentation of a group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises 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 ...
by
generators and
relations. It is much used in
geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ...
, the
fundamental group of a
simplicial complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
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
A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its ...
; and the classical
Burnside problem.
History
See the book by Chandler and Magnus for a detailed history of combinatorial group theory.
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
Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
, in the early 1880s, who gave the first systematic study of groups by generators and relations.
References
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