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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: cmay@towson.edu, jzimmerman@towson.edu
JAY ZIMMERMAN
Affiliation:
Department of Mathematics, Towson University, Baltimore, MD 21252, USA e-mail: cmay@towson.edu, jzimmerman@towson.edu
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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

REFERENCES

1.Besche, H. U., Eick, B. and O'Brien, E. A., A millennium project: Contructing small groups, Int. J. Algebra Comput. 12 (2002), 623–644.CrossRefGoogle Scholar
2.Blackburn, S. R., Neumann, P. M. and Venkataraman, G., Enumeration of finite groups (Cambridge University Press, Cambridge, UK, 2007).Google Scholar
3.Burnside, W., Theory of groups of finite order (Cambridge University Press, Cambridge, UK, 1911).Google Scholar
4.Conder, M. D. E. and Tucker, T. W., The symmetric genus spectrum of finite groups, ARS Math. Contemp. 4 (2011), 271–289.Google Scholar
5.Conway, J. H., Dietrich, H. and O'Brien, E. A., Counting groups: Gnus, moas and other exotica, Math. Intell. 30 (2008), 6–15.CrossRefGoogle Scholar
6.Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups, 4th Edition (Springer-Verlag, Berlin, 1957).Google Scholar
7.Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
8.Gross, J. L. and Tucker, T. W., Topological graph theory (John Wiley and Sons, New York, 1987).Google Scholar
9.May, C. L., The real genus of 2-groups, J. Algebra Appl. 6 (2007), 103–118.Google Scholar
10.May, C. L. and Zimmerman, J., Groups of small strong symmetric genus, J. Group Theory 3 (2000), 233–245.Google Scholar
11.May, C. L. and Zimmerman, J., Groups of symmetric genus σ ≤ 8, Comm. Algebra 36 (2008), 4078–4095.Google Scholar
12.May, C. L. and Zimmerman, J., The 2-groups of odd strong symmetric genus, J. Algebra Appl. 9 (2010), 465–481.Google Scholar
13.Miller, G. A., Determination of all the groups of order pm which contain the abelian group of type (m-2,1), p being any prime, Trans. Am. Math. Soc. 2 (1901), 259–272.Google Scholar
14.Miller, G. A., On the groups of order pm which contain operators of order p m−2, Trans. Am. Math. Soc. 3 (1902), 383–387.Google Scholar
15.Singerman, D., On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Phil. Soc. 76 (1974), 233–240.Google Scholar
16.Tucker, T. W., Finite groups acting on surfaces and the genus of a group, J. Comb. Theory Ser. B 34 (1983), 82–98.Google Scholar