On automorphisms of some $p$-groups

Abstract

A group $G$ is said to enjoy ``Hasse principle'' if every local coboundary of $G$ is a global coboundary. Let $G$ be a non-Abelian finite $p$-group of order $p^m$, $p$ prime and $m > 4$ having a normal cyclic subgroup of order $p^{m-2}$ but having no element of order $p^{m-1}$. We prove that $G$ enjoys ``Hasse principle'' if $p$ is odd but in the case $p = 2$, there are fourteen such groups twelve of which enjoy ``Hasse principle'' but the remaining two do not satisfy ``Hasse principle''. We also find all the conjugacy preserving outer automorphisms for these two groups.