Editing Combinatorial group theory

In [[mathematics]], '''combinatorial group theory''' is the theory of [[free group]]s, and the concept of a [[presentation of a group]] by [[generator (mathematics)|generators]] and [[relation (mathematics)|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]].

== History ==
See the book by Chandler and Magnus for a detailed history of combinatorial group theory.<ref>{{citation | title = The History of Combinatorial Group Theory: A Case Study in the History of Ideas
| series = Studies in the History of Mathematics and Physical Sciences
| first1 = B. | last1 = Chandler | first2 = Wilhelm | last2 = Magnus
| authorlink2 = Wilhelm Magnus
| publisher = Springer
| edition = 1st
| date = December 1, 1982
| isbn = 978-0-387-90749-9
}}</ref>

A proto-form is found in the 1856 [[icosian calculus]] of [[William Rowan Hamilton]], where he studied the [[icosahedral symmetry|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.<ref name="stillwell374">{{Citation
| publisher = Springer
| isbn = 978-0-387-95336-6
| last = Stillwell
| first = John
| title = Mathematics and its history
| date = 2002
| page = [https://books.google.com/books?id=WNjRrqTm62QC&pg=PA374 374]
}}</ref>

== References ==
{{reflist}}

[[Category:Combinatorial group theory| ]]