Elementary proof of a theorem of Blackburn

Abstract

For a positive integer n, a finite p-group G is called an M_n-group, if all subgroups of index pⁿ of G are metacyclic, but there is at least one subgroup of index p^n−1 that is not. A classical result in p-group theory is the classification of M₁-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result.

Key words： Finite p-group; M_n-group; minimal non-abelian group; minimal non-metacyclic group