Rendiconti del Seminario Matematico della Università di Padova


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

Published online: 2013-05-15

Automorfismi involutori di $p$-gruppi finiti

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

(1) Liceo Scientifico G.B. Benedetti, Venezia, Italy
(2) Università di Ca' Foscari, Mestre (Venezia), Italy
(3) Università di Ca' Foscari, 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