Group Presentation

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written <(x_i)_(i in I)|(r)_(r in R)>, where r=1 (the identity element) is often written in place of r. A group presentation defines the quotient group of the free group F(I) by the normal subgroup generated by R, which is the group generated by the generators x_i subject to the relations r in R.

Examples of group presentations include the following.

1. The presentation <x,y|x^2=1,y^n=1,(xy)^2=1> defines a group, isomorphic to the dihedral group D_n of finite order 2n, which is the group of symmetries of a regular n-gon.

2. The fundamental group of a surface of genus g has the presentation


3. Coxeter groups.

4. Braid groups.

See also

Presentation Matrix

This entry contributed by Yves de Cornulier

Explore with Wolfram|Alpha


Johnson, D. L. Presentations of Groups, 2nd ed. Cambridge, England: Cambridge University Press, 1997.Sims, C. C. Computation with Finitely Presented Groups. Cambridge, England: Cambridge University Press, 1994.Stillwell, J. Classical Topology and Combinatorial Group Theory, 2nd ed. New York: Springer-Verlag, 1993.

Referenced on Wolfram|Alpha

Group Presentation

Cite this as:

de Cornulier, Yves. "Group Presentation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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