REU 1994: Difference sets in groups (H. Pollatsek)

A finite group contains a difference set if and only if the group acts as a regular group of automorphisms of a symmetric design. Questions about difference sets sit, therefore, at the intersection of group theory, design theory and coding theory. Difference sets in abelian groups have been studied since the classic paper of Singer in 1938. Since 1990 interest has grown in non-abelian difference sets, and the non-abelian case demands additional tools from representation theory. Looking for difference sets in specific finite groups works well as a task for undergraduate researchers. There are a number of open existence questions in relatively small groups that lend themselves to a mixture of theoretical analysis and brute force computer search. (See for example the surveys [Davis and Jedwab] and [Smith] listed below.) Students in the 1994 REU program at Mount Holyoke obtained abelian and non-abelian results of considerable interest to researchers. Although the results are somewhat specialized, the methods range widely over finite group theory, representation theory, and algebraic number theory, affording the students an opportunity to learn and use some important mathematics.

Student participants:

The students worked on four problems. Two papers have appeared.   The first paper is  "The Star Theorem" by Martin. The  second paper is "Ruling Out (160, 54, 18) Difference Sets in Some Nonabelian Groups" by Alexander, Balasubramanian, Martin, Monahan, Pollatsek and Sen, in the Journal of Combinatorial Designs, 8: 221-231, 2000.
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 X Z2 (for example if G is isomorphic to D20 X 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, the Frattini subgroup of S has order 8, and one of the following is true: (a) G = S X 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 A Gamma L(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 of the elementary abelian quotient of order 32 give a contradiction.  In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's "dihedral trick."  Theorem B summarizes the open onabelian cases based on this work.


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