Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich:

Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is not an inner one. ($p$ as usual denotes a prime number)

I would like to know if anybody knows anything concerning history of this problem.

What I know are as follows: W. Gaschütz has proved in 1966 that every finite $p$-group of order greater than $p$ has a non-inner automorphism of $p$-power order. One year before Gaschütz, H. Liebeck has proved problem 4.13 for $p$-groups of class $2$ whenever $p>2$. The problem in its own is of course interesting, but I think maybe the main motivation to propose the problem is to strengthen Gaschütz's famous result, am I right?

What I really look for is to know: does the problem come from other mathematics problems that people already noted to and they stated it? For example, has it any relation to the classification of finite simple groups?!

I am sorry if you feel my questions are so vague! and I apologize in advance for.

share|improve this question
You mean Wolfgang Gaschütz, don't you? Please do him the honour of spelling his name correctly. –  Johannes Hahn Dec 21 '11 at 18:26
Yes, Wolfgang Gaschütz is meant. I edited the question accordingly. –  Max Horn Dec 21 '11 at 22:19
add comment

1 Answer

up vote 2 down vote accepted

One relevant recent direction is given in my 2007 paper with Geir Helleloid, The automorphism group of a finite p-group is almost always a p-group. J ALGEBRA vol. 312, (1) 294-329. http://www.sciencedirect.com/science/article/pii/S0021869307000142 also at http://arxiv.org/abs/math/0602039

We show 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 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.

share|improve this answer
I don't know if there is a relation between classification of simple groups and existence of a non-inner automorphism of order p. Have you found some relation in the last year? –  Marco Ruscitti Oct 29 '13 at 9:35
@Marco: not yet! –  Alireza Abdollahi Oct 29 '13 at 12:18
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.