Automorfismi involutori di $p$-gruppi finiti

Abstract

Let $p$ be a fixed odd prime number. In this note we study the class of finite $p$-groups $G$ admitting an automorphism $\phi$ of order $2$ such that $G=⟨g^{-1}{g}^{\phi }\mid g\in G⟩$ and $(g^{-1}{g}^{\phi }{)}^p=1$ for all $g\in G$. In this paper we prove that if the derived length of $G$ is $d$ and $C_G(\phi )$ is nilpotent of class $c$, then the nilpotency class of $G$ is bounded by a function depending only on $d$, $c$ and $p$. We prove also that if $p=3$ and $C_G(\phi )$ is nilpotent of class $c$, then $G$ is nilpotent of class at most $2c+1$.