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 nonabelian difference sets, and the nonabelian 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 nonabelian 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:

Jason Alexander '95, Lewis and Clark College

Rajalakshmi Balasubramanian '96, Mount Holyoke College

Jeremy Martin '96, Harvard University

Kimberley Monahan '95, College of the Holy Cross

Ashna Sen '96, Mount Holyoke College
The students worked on four problems.

They showed that the dihedral group of order 70 cannot contain a difference
set. (Subsequently we discovered this result was essentially in the literature.)

They proved that the direct product of the dihedral group D20 of order
20 and the elementary abelian group of order 8 cannot contain a difference
set with parameters (160, 54, 18), the parameters of a recently discovered
symmetric design.

They created a search algorithm to discover whether a group of order 400
having the elementary abelian group of order 25 as a homomorphic image
can contain a Hadamard difference set. Proving nonexistence would answer
several open questions about the existence of Hadamard difference sets.
[Unfortunately, although the algorithm is correct, the computer implementation
of the search procedure was flawed.]

Martin proved a theorem characterizing relations among powers of a primitive
root of unity z in the ring Z[z].
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: 221231, 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 5Sylow subgroup and
an elementary abelian 2Sylow 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 2Sylow subgroup
S and 5Sylow 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.
References

J.A. Davis and J. Jedwab, A survey of Hadamard difference sets, preprint,
1994.

D. Jungnickel, "Difference Sets" in Contemporary Design Theory: A Collection
of Surveys, J.H. Dinitz and D.R. Stinson eds., John Wiley and Sons, 1992.

E.S. Lander, Symmetric Designs: an algebraic approach, London Math. Soc.
Lecture Note Series 74, Cambridge University Press, 1983.

K.W. Smith, On Extending Lander's Table of Difference Sets: Searching for
nonabelian difference sets, unpublished, 1992.
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