A group presentation is a description of a set of generators and a subset of the free group generated by . It is written , where (the identity element) is often written in place of . A group presentation defines the quotient group of the free group by the normal subgroup generated by , which is the group generated by the generators subject to the relations .

Examples of group presentations include the following.

1. The presentation defines a group, isomorphic to the dihedral group of finite group order , which is the group of symmetries of a regular polygon with sides.

2. The fundamental group of a closed connected orientable surface of genus has the presentation

3. Coxeter groups.

4. Braid groups.

See also

Artin Group, Group Equation, Local Group Presentation, Normal Closure

Portions of this entry contributed by Yves de Cornulier

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References

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.

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Group Presentation

Cite this as:

Weisstein, Eric W., with contributions by Yves de Cornulier. "Group Presentation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GroupPresentation.html

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