### DIMACS Special Seminar

Title: The Automorphism Group of a Finite p-Group is Almost Always a p-Group

Speaker: **Geir Helleloid**, Stanford University

Date: Tuesday, February 20, 2007 9:30 am

Location: DIMACS Center, CoRE Bldg, Room 433, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Many common finite p-groups admit automorphisms of order coprime to p,
and when p is odd, it is reasonably difficult to find finite p-groups
whose automorphism group is a p-group. It turns out, however, that
the automorphism group of a finite p-group is almost always a
p-group. The asymptotics in the theorem involve fixing any two of the
following parameters and letting the third go to infinity: the
Frattini length, the number of generators, and p. The proof of this
theorem depends on a variety of topics: counting subgroups of a
p-group, analyzing the Frattini series of a free group via its
connection with the free Lie algebra, counting submodules of a module
via Hall polynomials, and using numerical estimates on Gaussian
coefficients.