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European Mathematical Society Publishing House
2016-09-19 17:05:49
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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129
2013
0
Automorfismi involutori di $p$-gruppi finiti
Egle
Bettio
Liceo Scientifico G.B. Benedetti, VENEZIA, ITALY
Giorgio
Busetto
Università di Ca' Foscari, MESTRE (VENEZIA), ITALY
Enrico
Jabara
Università di Ca' Foscari, VENEZIA, ITALY
Finite p-groups, involutive automorphisms, Brick loops
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 $2c+1$.
Group theory and generalizations
Combinatorics
General
1
15
10.4171/RSMUP/129-1
http://www.ems-ph.org/doi/10.4171/RSMUP/129-1