2 F eb 2 00 6 Almost All p - Groups Have Automorphism Group a p - Group When p is Odd
نویسنده
چکیده
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 with automorphism group a p-group. Yet the goal of this paper is to prove that almost all finite p-groups do have automorphism group a p-group when p is odd. The asymptotic sense in which the theorem holds involves bounding the Frattini length of the p-groups and letting the number of generators go to infinity. The proof of the theorem draws on a detailed analysis of the Frattini series of a free group and the combinatorics linking finite p-groups and representations of GL(n, Fp). The case of p = 2 remains open.
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