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THE SYMMETRIC GENUS OF 2-GROUPS

Published online by Cambridge University Press:  02 August 2012

COY L. MAY  and
JAY ZIMMERMAN
COY L. MAY
Affiliation:
Department of Mathematics, Towson University, Baltimore, MD 21252, USA e-mail: [email protected], [email protected]
JAY ZIMMERMAN
Affiliation:
Department of Mathematics, Towson University, Baltimore, MD 21252, USA e-mail: [email protected], [email protected]
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Abstract

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Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

Keywords

Primary 20F38Secondary 20D1520H1030F9957M60
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 9 - 21
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

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