Groups of operators that fix a Sylow subgroup
Let \(A\) and \(G\) be finite groups and suppose that \(A\) acts on \(G\). \(A\) is said to be an \(S\)-group of operators of \(G\) if for each prime number \(r\) such that \(r\mid|G|\), \(G\) has a unique \(A\)-invariant \(r\)-Sylow subgroup. The group \(G\) is said to be \(A\)-\(p\)-nilpotent if \(A\) is an \(S\)-group and \(|G|=p^n\), \(p\) a prime number. \textit{A. Beltrán} proved [in Arch. Math. 69, No. 3, 177-184 (1997; Zbl 0917.20005)] that if \((|G|,|A|)=1\) and \(C_G(A)\) is a nilpotent group, then \(G\) is a soluble group. The author proves that the \(A\)-\(p\)-nilpotent groups are \(p\)-nilpotent and soluble groups. He also obtains that the Fitting length of \(G\) is bounded by \(n+2\) and gives some counter-example to show that the bound cannot be improved.
scientific article; zbMATH DE number 1993738
scientific article