The group of automorphisms of a class of finite 𝑝-groups

Abstract

Let G G be a finite p p -group and denote by K i ( G ) {K_i}(G) the members of the lower central series of G G . We call G G of type ( m , n ) (m,\,n) if (a) G G has nilpotency class m − 1 m - 1 , (b) G / K 2 ( G ) ≅ Z p n × Z p n G/{K_2}(G) \cong {{\mathbf {Z}}_{{p^n}}} \times {{\mathbf {Z}}_{{p^n}}} and K i ( G ) / K i + 1 ( G ) ≅ Z p n {K_i}(G)/{K_{i + 1}}(G) \cong {{\mathbf {Z}}_{{p^n}}} for every i i , 2 ⩽ i ⩽ n − 1 2 \leqslant i \leqslant n - 1 . In this work we describe the structure of Aut ⁡ ( G ) \operatorname {Aut} (G) and certain relations between Out ⁡ ( G ) \operatorname {Out} (G) and G G .