Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism

Authors:
Tao Feng
Affiliations:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

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Abstract

Let p be a prime larger than 3 and congruent to 3 modulo 4, and let G be the non-abelian group of order p^3 and exponent p. We study the structure of a putative difference set with parameters (p^3,p^3-12,p^3-34) in G which is fixed by a certain element of order p in Aut(G). We then give a construction of skew Hadamard difference set in the group G for each prime p3 that is congruent to 3 modulo 4. This is the first infinite family of non-abelian skew Hadamard difference sets. Finally, we show that the symmetric designs derived from these new difference sets are not isomorphic to the Paley designs.