Geodesic
THE SYMMETRIC GENUS OF 2-GROUPS
Glasgow mathematical journal, Tome 55 (2013) no. 1, pp. 9-21

Voir la notice de l'article provenant de la source Cambridge University Press

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).
DOI : 10.1017/S0017089512000316
Mots-clés : Primary 20F38, Secondary 20D15, 20H10, 30F99, 57M60 
MAY, COY L.; ZIMMERMAN, JAY. THE SYMMETRIC GENUS OF 2-GROUPS. Glasgow mathematical journal, Tome 55 (2013) no. 1, pp. 9-21. doi: 10.1017/S0017089512000316
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[1] 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. Google Scholar | DOI

[2] 2.Blackburn, S. R., Neumann, P. M. and Venkataraman, G., Enumeration of finite groups (Cambridge University Press, Cambridge, UK, 2007). Google Scholar

[3] 3.Burnside, W., Theory of groups of finite order (Cambridge University Press, Cambridge, UK, 1911). Google Scholar

[4] 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] 5.Conway, J. H., Dietrich, H. and O'Brien, E. A., Counting groups: Gnus, moas and other exotica, Math. Intell. 30 (2008), 6–15. Google Scholar | DOI

[6] 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] 7.Gorenstein, D., Finite groups (Harper and Row, New York, 1968). Google Scholar

[8] 8.Gross, J. L. and Tucker, T. W., Topological graph theory (John Wiley and Sons, New York, 1987). Google Scholar

[9] 9.May, C. L., The real genus of 2-groups, J. Algebra Appl. 6 (2007), 103–118. Google Scholar

[10] 10.May, C. L. and Zimmerman, J., Groups of small strong symmetric genus, J. Group Theory 3 (2000), 233–245. Google Scholar

[11] 11.May, C. L. and Zimmerman, J., Groups of symmetric genus σ ≤ 8, Comm. Algebra 36 (2008), 4078–4095. Google Scholar

[12] 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] 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] 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] 15.Singerman, D., On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Phil. Soc. 76 (1974), 233–240. Google Scholar

[16] 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

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