Next Meeting

Topics:

An Algorithm to Express Words as a Product of Conjugates of Relators Ellen Ziliak Colorado State University In this talk I will describe an algorithm which uses a subgraph of the full Cayley Graph to express words as a product of conjugates of relators. To begin, I will describe how this algorithm is useful for computing in group extensions. Next I will describe the subgraph of the full Cayley Graph which is called a partial automaton. We will see how this subgraph can be used to construct a free generating set for a subgroup of the free group. After this construction we will see how the partial automaton can be used to actually do the rewriting. In the end we will discuss some storage issues.

Storer's Conjecture and Nonabelian Hadamard Difference Sets Ken Smith Sam Houston State University, Huntsville, Texas Difference sets in finite groups correspond to symmetric designs with a transitive automorphism group. These combinatorial objects have applications to statistics, computer technology, signal processing and cryptology. The study of these objects interweaves number theory, combinatorics and group theory. Difference sets in cyclic groups were studied as early as the 1930's with Singer's construction using the multiplicative group of a finite field. In the 1970s, difference sets in finite groups (some nonabelian) were studied by mathematicians at the National Security Agency. Should one study difference sets in nonabelian groups? A folklore conjecture (attributed to Tom Storer) suggested that any difference set occurring in a nonabelian group implied the existence of one with the same parameters in an abelian group. On the other hand, Robert Liebler (Colorado State University) argued that most interesting combinatorial structures had nonabelian automorphism groups and so nonabelian groups are the natural environment for combinatorial objects, including difference sets. In this talk we introduce the theory of difference sets and trace the work that led to the first counterexample to Storer's conjecture. This talk is dedicated to the memory of Robert Liebler (1944-2009) whose ideas motivated much of this research.