nLab finitely presentable group

Finitely presented groups

Finitely presented groups


A group GG is finitely presentable if it has a finite presentation, i.e., there is a group presentation, X:R\langle X: R\rangle, for GG with both its set, XX, of generators and its set, RR, of relations being finite sets.


The term ‘finitely presented’ is often used rather than `finitely presentable', however 'finitely presented' would seem to imply that a given finite presentation was intended, whilst here only the existence of one is required.


Textbook accounts on group presentations:

Lecture notes:

  • Derek Holt (notes by Florian Bouyer), Presentation of Groups (2013) [pdf]

On (finitely) presented groups as fundamental groups of (finite) simplicial complexes/CW-complexes:

  • Joseph J. Rotman, around Thm. 7.34 in: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119 (1988) [[doi:10.1007/978-1-4612-4576-6]]

  • Behrooz Mashayekhy, Hanieh Mirebrahimi, Some Properties of Finitely Presented Groups with Topological Viewpoints, International Journal of Mathematics, Game Theory and Algebra 18 6 (2010) 511-515 [arXiv:1012.1744]

Last revised on February 3, 2023 at 07:55:26. See the history of this page for a list of all contributions to it.