Ruling out (160, 54, 18) difference sets in some nonabelian groups

Abstract

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D20 × Z2 (for example, if G ' D20 ×K). (2) G has a normal 5–Sylow subgroup and an elementary abelian 2–Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5 or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2–Sylow subgroup S and 5–Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |φ(S)| = 8, and one of the following is true: (a) G = S ×T and S is nonabelian; (b) G has a D10 image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in AΓL(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations ∗Support is gratefully acknowledged from the National Science Foundation Research Experiences for Undergraduates program, the Pew Charitable Trusts via the New England Consortium for Undergraduate Science Education, and the Howard Hughes Medical Foundation. Most of this work was done under Harriet Pollatsek’s supervision in the summer 1994 undergraduate mathematics research institute at Mount Holyoke College. The other authors are the (then) undergraduate researchers: Jason Alexander ’95, Lewis and Clark College, in the doctoral program in the philosophy and foundations of mathematics at UC Irvine; Rajalakshmi Balasubramanian ’96, Mount Holyoke College, in the doctoral program in statistics at Harvard University; Jeremy Martin ’96, Harvard University, in the doctoral program in mathematics at UC San Diego; Kimberley Monahan ’95, College of the Holy Cross, now teaching high school mathematics; Ashna Sen ’96, Mount Holyoke College, who completed a master’s degree in geophysics at Stanford University. 1 Ruling out (160,54,18) difference sets in some nonabelian groups (with Jason Alexander, Rajalakshmi Balasubramanian, Kimberly Monahan, Harriet Pollatsek, and Ashna Rubina Sen) J. Combin. Designs 8 (2000), no. 4, 221--231. Publisher's official version: http://dx.doi.org/10.1002/1520-6610(2000)8:4 3.0.CO;2-6, Open Access version: http://kuscholarworks.ku.edu/dspace/. [This document contains the author's accepted manuscript. For the publisher's version, see the link in the header of this document.]