Abstract
Let \({\mathcal {S}}\) denote the set of positive integers that may appear as the strong symmetric genus of a finite abelian group. We obtain a set of (simple) necessary and sufficient conditions for an integer g to belong to \({\mathcal {S}}\). We also prove that the set \({\mathcal {S}}\) has an asymptotic density and approximate its value.
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References
B. Borror, A. Morris, and M. Tarr, The strong symmetric genus spectrum of abelian groups, Rose-Hulman Undergrad. Math. J. 15 (2014), 115–130.
J.L. Gross and T.W. Tucker, Topological Graph Theory, Wiley, New York, 1987.
H. Halberstam and K.F. Roth, Sequences, Oxford University Press, Oxford, 1966.
A. Hurwitz, Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–442.
C. Maclachlan, Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15 (1965), 699–712.
C.L. May and J. Zimmerman, There is a group of every strong symmetric genus, Bull. London Math. Soc. 35 (2003), 433–439.
H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge University Press, Cambridge, 2007.
K. Prachar, Über die kleinste quadratfreie Zahl einer arithmetischen Reihe, Monatsh. Math. 62 (1958), 173–176.
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Kumchev, A.V., May, C.L. & Zimmerman, J.J. The strong symmetric genus spectrum of abelian groups. Arch. Math. 108, 341–350 (2017). https://doi.org/10.1007/s00013-016-1018-8
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DOI: https://doi.org/10.1007/s00013-016-1018-8