Short proofs of some basic characterization theorems of finite p-group theory

Abstract

We oer short proofs of such basic results of nite p-group theory as theorems of Blackburn, Huppert, Ito-Ohara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic p-group and R is a proper G-invariant subgroup of G0, then G=R is not metacyclic. In the second part we use Blackburn's theory of p-groups of maximal class. Here we prove that a p-group G is of maximal class if and only if 2 (G) = hx 2 G j o(x) = p 2i is of maximal class. We also show that a noncyclic p-group G of exponent > p contains two distinct maximal cyclic subgroups A and B of orders > p such that jA Bj = p, unless p = 2 and G is dihedral.