The automorphism group of a finite p-group is almost always a p-group

Geir T. Helleloid, Ursula Martin

Research output: Contribution to journalArticlepeer-review


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. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the third go to infinity: the lower p-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 lower p-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.
Original languageEnglish
Pages (from-to)294-329
Number of pages36
JournalJournal of Algebra
Issue number1
Publication statusPublished - 2007


  • p-Group
  • Automorphism group
  • Lower p-series
  • Frattini subgroup
  • Lower central p-series


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