Coprime actions with supersolvable fixed-point groups. (English) Zbl 1426.20010
All groups considered in this review are finite. In this paper, the authors consider elementary abelian \(r\)-groups, \(r\) a prime, acting on an \(r'\)-group \(G\), and suppose that the fixed-point group \(\text{C}_G(a)\) is supersoluble for all \(a\in A\setminus\{1\}\). They prove in Theorem 1.1 that if the order of \(A\) is at least \(r^4\), then \(G\) is supersoluble. This does not necessarily happen if \(| A|=r^3\). If \(A\) has order at least \(r^3\), according to Theorem 1.2, \(G'\leq \text{F}_3(G)\), the third term of the Fitting series of \(G\), and if either \(\text{C}_G(a)'\) is of odd order for each \(a\in A\setminus\{1\}\) or \(r\) is not a half-Fermat prime (that is, \(r\) is not of the form \((p^m+1)/2\) for a prime \(p\) and a natural \(m\)), then \(G'\leq \text{F}_2(G)\). The proofs of both theorems depend on the fact that, if \(A\) is an elementary abelian \(r\) group of order at least \(r^3\) acting on an \(r'\)-group \(G\) and \(p\) is a prime such that \(\text{C}_G(a)\) is \(p\)-nilpotent for every \(a\in A\setminus\{1\}\), then \(G\) is \(p\)-nilpotent (Theorem 1.3).
Reviewer: Ramón Esteban-Romero (València)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D45 | Automorphisms of abstract finite groups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
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