graph product of groups

Let Γ be a finite undirected graph and let {Gv:vV(Γ)} be a collectionMathworldPlanetmath of groups associated with the vertices of Γ. Then the graph productMathworldPlanetmath of the groups Gv is the group G=F/R, where F is the free product of the Gv and R is generated by the relationsMathworldPlanetmathPlanetmath that elements of Gu commute with elements of Gv whenever u and v are adjacent in Γ.

The free product and the direct productPlanetmathPlanetmathPlanetmath are the extreme examples of the graph product. To obtain the free product, let Γ be an anticlique, and to obtain the direct product, let Γ be a clique.


  • 1 E.R. Green, Graph products of groups, Doctoral thesis, The University of Leeds, 1990.
  • 2 S. Hermiller and J. Meier, Algorithms and geometry for graph products of groups, Journal of Algebra 117 (1995), 230–257.
  • 3 M. Lohrey and G. Sénizergues, When is a graph product of groups virtually-free?, to appear in Communications in Algebra. 2006 preprint available online at
  • 4 R.Brown, M. Bullejos, and T. Porter,‘Crossed complexes, free crossed resolutions and graph products of groups’, Proceedings Workshop Korea 2000, J. Mennicke, Moo Ha Woo (eds.) Recent Advances in Group Theory, Heldermann Verlag Research and Exposition in Mathematics 27 (2002) 8–23. arXiv:math/0101220
Title graph product of groups
Canonical name GraphProductOfGroups
Date of creation 2013-03-22 16:10:36
Last modified on 2013-03-22 16:10:36
Owner mps (409)
Last modified by mps (409)
Numerical id 8
Author mps (409)
Entry type Definition
Classification msc 20F65
Defines graph product of groups
Defines graph product