REU 2000: 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 [Jungnickel] and [Smith] listed below.)
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:

Juliana Belding '01, Bryn Mawr College

Elizabeth Carver '01, Georgia Southern University

Umut Gursel '01, Mount Holyoke College

Peter Johnston '01, Colorado College

Richard Lassow '01, St. Olaf College

Jenna Nichols '01, Mount Holyoke College

Laura Steuble '01, Mount Holyoke College
The students' project for summer 2000 was conceived and directed in collaboration
with Ken Smith of Central Michigan University. The students
investigated the possible existence of a difference set with parameters
(288, 42, 6). No difference sets or symmetric designs with these
parameters are known. However, Ken Smith has observed that these
parameters are compatible with the existence of a difference list
in a homomorphic image of order 72 obtained from a known difference set
with parameters (36, 15, 6). The students tested a construction method
based on Smith's observation, using it to construct 272 (96, 20,
4) difference sets from difference lists in homomorphic images of order
32 obtained from (16, 6, 2) difference sets. Work remains to determine
equivalencies among these and with known (96, 20, 4) difference sets, as
well as to determine equivalencies among the associated symmetric designs.
The students also determined numerous necessary conditions (intersection
numbers) for existence of a (288, 42, 6) difference set and obtained partial
results toward a construction from a difference list.
Jenna Nichols wrote an honors thesis ``Search for a (288, 42, 6) Difference
Set." In her thesis, she summarized and somewhat extended some of the summer's
work. In addition to what appears in the thesis, the students proved
that a group of order 288 containing a difference set cannot have a quotient
isomorphic to Z_{32} or (Z_2)^5. Intersection numbers were
also determined for all (including nonabelian) quotients of order
8 and all abelian quotients of exponent 2 or 4. A summary of the
summer's results is available electronically. Summary
A copy of the thesis is also available electronically: Thesis
NOTE (summer 2002): Students of Emily Moore at Grinnell
College have extended this work on (96, 20, 4) difference sets. They
constructed an additional family of difference sets from another (16, 6,
2) difference set, and they determined equivalencies among all the difference
sets constructed from (16, 6, 2) difference sets in this way. They
are continuing to work on the equivalence question for the associated symmetric
designs.
References

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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