nLab finitely presentable group

Idea

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

Remark

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.

Related concepts

• finitely generated group

References

Textbook accounts on group presentations:

• David Lawrence Johnson, Topics in Theory of Group Presentations, London Mathematical Society Lecture notes series 42, Cambridge Univ. Press (1980) [doi:10.1017/CBO9780511629303]

• David Lawrence Johnson, Presentations of Groups, London Mathematical Society Student Texts 15, Cambridge Univ. Press (1990) [doi:10.1017/CBO9781139168410]

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]