Rendiconti del Seminario Matematico della Università di Padova

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Volume 129, 2013, pp. 1–15
DOI: 10.4171/RSMUP/129-1

Automorfismi involutori di $p$-gruppi finiti

Egle Bettio[1], Giorgio Busetto[2] and Enrico Jabara[3]

(1) Liceo Scientifico G.B. Benedetti, Castello 2835, 30122, Venezia, Italy
(2) Dipartimento di Informatica, Università di Ca' Foscari, Via Torino 153, 30172, Mestre, Italy
(3) Dipartimento di Matematica Applicata, Università di Ca' Foscari, Dorsoduro 3825/e, 30123, Venezia, Italy

Let $p$ be a fixed odd prime number. In this note we study the class of finite $p$-groups $G$ admitting an automorphism ${\varphi} $ of order $2$ such that $G=\langle \, g^{-1}g^{{\varphi} } \mid g \in G \, \rangle $ and $(\kern 0.5pt g^{-1}g^{{\varphi} })^{\kern 1pt p}=1$ for all $g \in G$. In this paper we prove that if the derived length of $G$ is $d$ and $C_{G}({\varphi} )$ 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 $C_{G}({\varphi} )$ is nilpotent of class $c$, then $G$ is nilpotent of class at most $2c+1$.

Keywords: Finite p-groups, involutive automorphisms, Brick loops

Bettio Egle, Busetto Giorgio, Jabara Enrico: Automorfismi involutori di $p$-gruppi finiti. Rend. Sem. Mat. Univ. Padova 129 (2013), 1-15. doi: 10.4171/RSMUP/129-1