About finite solvable groups with exactly four \(p\)-regular conjugacy classes. (English) Zbl 0854.20022
The finite groups with three \(p\)-regular classes were considered by Y. Ninomiya [Arch. Math. 57, No. 2, 105-108 (1991; Zbl 0774.20007), Can. J. Math. 43, No. 3, 559-579 (1991; Zbl 0738.20012) and 45, No. 3, 626 (1993; Zbl 0826.20011)]. Let \(p\) be an odd prime. The main result of the paper under review is the classification of the soluble groups \(G\) with \(O_p(G)=1\) and with exactly four \(p\)-regular classes of conjugate elements. Their list is too long to be reproduced here; a common property of these groups is that they have \(p\)-length 1.
Reviewer: M.Deaconescu (Safat)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
