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 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 [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:

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.


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