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On central difference sets in certain non-abelian 2-groups. (English) Zbl 1089.05014

Summary: We define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group \(G\) of Suzuki type with \(|G|=2^{2s}\) always possesses a non-trivial difference set. We show that if \(s\) is odd, \(G\) possesses a central difference set, whereas if \(s\) is even, \(G\) has no non-trivial central difference set.

MSC:

05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)

Keywords:

conjugacy class
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References:

[1] Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1999), Cambridge University Press: Cambridge University Press Cambridge
[2] Dillon, J. F., Variations on a scheme of McFarland for noncyclic difference sets, J. Combin. Theory Ser. A, 40, 9-21 (1985) · Zbl 0583.05016
[3] Huppert, B.; Blackburn, N., Finite Groups II (1982), Springer: Springer Berlin, Heidelberg, New York · Zbl 0477.20001
[4] R.A. Liebler, Constructive representation theoretic methods and non-abelian difference sets, in: A. Pott, P.V. Kumar, T. Helleseth, D. Jungnickel (Eds.), Difference Sets, Sequences and their Correlation Properties (Bad Winsheim, 1998), NATO Advanced Science Institute Series C: Mathematical Physics Science, vol. 542, Kluwer Academic Publishers, Dordrecht, 1999, pp. 331-352.; R.A. Liebler, Constructive representation theoretic methods and non-abelian difference sets, in: A. Pott, P.V. Kumar, T. Helleseth, D. Jungnickel (Eds.), Difference Sets, Sequences and their Correlation Properties (Bad Winsheim, 1998), NATO Advanced Science Institute Series C: Mathematical Physics Science, vol. 542, Kluwer Academic Publishers, Dordrecht, 1999, pp. 331-352. · Zbl 0940.05017
[5] Mann, H. B., Difference sets in elementary abelian groups, Illinois J. Math., 9, 212-217 (1965) · Zbl 0135.06103
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