Replacement Of The Simple Group Psl (3, P)

The orbits of the point stabilzer of a finitc permutation group are called the sub-orbits of the permutation group. The determination of the suborbits of a permutation group is one of the basic problems in the permutation group theory. It plays an important role in the study of combinatorial structures. However, such a problem is very difficult in general, and it depends heavily on the understanding of abstract groups and the application of combinatorial tools as well. Recently, a lot of results have been apperaed dealing with the simple groups PSL(2,p). As for the simple groups PSL(3,p), such results are quite rare. Under some conditions, the group PSL(3,p) contains the three smallest simple groups, that is, A₅,PSL{2,7), A₆. Therefore, it is very interesting to determine the suborbit structure of PSL(3,p) reletive to these three simple groups. Moreover, it will play an important role in the study of the symmetry of combinatorial stuctures. The start point of this thesis is the determination of maximal subgroups of PSL(3, p^k). In this thesis, we consider the action of PSL(3,p) on the set of right cosets of the simple subgroup A₅ by the right multiplication, and determine the suborbits of this simple group, while it is assumed that p ≡ l(mod 60), for a convenience, and the other cases can be consided similarly.