Frobenius-like groups as groups of automorphisms. (English) Zbl 1321.20023
A Frobenius-like group is a finite group which is a product \(FH\) of two subgroups \(F\) and \(H\) where \(F\) is a non-trivial nilpotent normal subgroup of \(FH\), \(H\) is not trivial, and \(FH/[F,F]\) is a Frobenius group with Frobenius kernel \(F/[F,F]\). When such a group acts on a finite group \(G\), the structure of \(G\) and that of \(C_G(H)\) are related. A number of papers have been written recently describing such relationship with a special focus on bounds on the Fitting length and related parameters. In the present paper, the authors survey these results. In addition, they prove a new result which generalizes to some Frobenius-like groups one of the results in [E. I. Khukhro and N. Yu. Makarenko, J. Algebra 386, 77-104 (2013; Zbl 1293.20020)].
Reviewer: Alexandre Turull (Gainesville)
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20D30 | Series and lattices of subgroups |
