A characterization of some \(\mathrm{PGL}(2,q)\) by maximum element orders. (English) Zbl 1333.20012
Summary: In this paper, we characterize some \(\mathrm{PGL}(2,q)\) by their orders and maximum element orders. We also prove that \(\mathrm{PGL}(2,p)\) with \(p\geqslant 3\) a prime can be determined by their orders and maximum element orders. Moreover, we show that, in general, if \(q=p^n\) with \(p\) a prime and \(n>1\), \(\mathrm{PGL}(2,q)\) can not be uniquely determined by their orders and maximum element orders. Several known results are generalized.
MSC:
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
