Since I started to work in the field of finite p-groups I have encountered, sometimes to my surprise, results concerned with the number of generators, and gradually I became convinced that this is an area worth investigating in its own right, with applications to other areas (such as padic Lie groups or Schur multipliers). In the present paper I try to collect some of the results about generators that seem to me to be the most interesting, admitting a natural bias towards my own work. This being a survey article, proofs are not usually given,except when not available elsewhere or as an illustration.

Some notation. The word “group” usually means a finite p-group, cl G and exp G are the class and exponent of G, d(G) and r(G) are the minimal numbers of generators and relations of G, G', Gi, ∏i(G), ф(G), Z(G), M(G) are the commutator subgroup, the i-th term of the lower central series, the subgroup generated by p -th powers, the Frattini subgroup,the centre and the Schur multiplier, Cn is a cyclic group of order n, wr stands for wreath product. Also, x is the smallest integer not less than the real number x, and logarithms are always to the base.

NUMBER OF GENERATORS

The simplest restriction is, of course, just to assume that dG) is given (or bounded). This is a very weak assumption. Indeed, any p-group can be embedded in a 2-generator one [NN]. Still, we mention two deep results. The first is Kostrikin's, stating that there are only finitely many p-groups of exponent p with a given number of generators.