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

Abstract:

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