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The symmetric crosscap number of a group

Published online by Cambridge University Press:  25 July 2002

Coy L. May
Affiliation:
Department of Mathematics, Towson University, Baltimore, Maryland 21252, USA e-mail: cmay@towson.edu
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Let G be a finite group. The symmetric crosscap number\tilde \sigma (G) is the minimum topological genus of any compact non-orientable surface (with empty boundary) on which G acts effectively. We first survey some of the basic facts about the symmetric crosscap number; this includes relationships between this parameter and others. We obtain formulas for the symmetric crosscap number for three families of groups, the dicyclic groups, the abelian groups with most factors in the canonical form isomorphic to Z_(2), and the hamiltonian groups with no odd order part. We also determine \tilde \sigma (G) for each group G with order less than 16. The groups with symmetric crosscap numbers 1 and 2 have been classified. We show here that there are no groups with \tilde\sigma=3; this affirms a conjecture of Tucker.

Type
Research Article
Copyright
2001 Glasgow Mathematical Journal Trust