Finite groups whose conjugacy class graphs have few vertices. (English) Zbl 1104.20027
In this note, the finite groups are classified satisfying the following property \(P_5\): their conjugacy class lengths are set-wise relatively prime for any 5 distinct classes. As such, it extends results on finite groups satisfying the analogous property \(P_3\), as done by M. Fang and P. Zhang [J. Algebra 264, No. 2, 613-619 (2003; Zbl 1024.20021)]. The list of all the finite groups satisfying \(P_5\), comprises the contents of Theorem A of the paper; there are 31 isomorphy types of those groups being non-Abelian besides all the Abelian groups.
Reviewer: R. W. van der Waall (Huizen)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20E45 | Conjugacy classes for groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D05 | Finite simple groups and their classification |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
