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## An "algebraic K-theory" of finite groups: The unstable case and reclusive primes

### Keith Dennis

Cornell University

###
Thursday, May 17, 2007

007 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: ** Persi Diaconis asked me if it might be possible
to develop a "K-theory of finite groups" based on n-tuples of
generating sets, which would replace rows of invertible matrices (the
bases of free modules). I found a way to construct groups that have a
universal mapping property analogous to those of free modules: A group
H is "n-homogeneous" if for every pair of ordered generating n-tuples,
there exists a unique automorphism of the group which takes the first
to the second.

It turns out that such groups were discovered by
B. H. Neumann & H. Neumann in 1951. Given a finite group G generated
by r elements, there is a natural way to construct an n-homogeneous
group H(n,G) for n>r. The automorphism groups of these groups play
the role of the group of invertible matrices in K-theory, and one
develops the stable algebra of these groups as n goes to infinity, as
in traditional algebraic K-theory.

Recently I have been looking at
the unstable case, rather than the stable case arising in K-theory.
Really interesting things happen when r is the minimum number of
generators of the group G.

The general construction gives a group
H(r,G) that has r generators all of the same order. For n>r, it is
trivial to see that this number is the exponent of G. The question
then arises: what is this number for n=r? After trying unsuccessfully
to prove that it was the exponent, I sought counterexamples, and to my
surprise, I found them---surprisingly, so far all have turned out to
be Frobenius groups. These considerations lead to an invariant that
generalizes the classical Scharlau invariant. There are many
interesting questions remaining even in this simplest case.

This talk will be accessible to graduate students.