Question

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.

Answer 1

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.