A Class Of Primitive Groups Of The Track Structure,

Suppose that G is a permutation group on the finite set Ω ,. Let G_α= {g ∈ G | α~g = α} be the point stabilizer of G, for any α ∈Ω Then the orbits of G_a on Ω are called the suborbits of G relative to α, while {α} is said to be trivial. A nonempty subset A is called a block of G if △~9 = △ or △~9 ∩ △ = θ, (?)_g ∈ G. Clearly every subset containing only one point and Ω itself is a block, called trivial blocks. The group G is said to be primitive if there exists no nontrivial block on Ω. It is well-known that G is primitive if and only if G_α is maxmial in G for any α ∈Ω .The determination of the suborbits of a permutation group is one of 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, even for primitive groups 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. In this thesis, we consider the action of PSL(3, p) on the set of right cosets of a maxiaml subgroup PSL(2,7) by the right multiplication, and determine the suborbits of this primitive group, while it is assumed that p = l(mod 168), for a convenience. The start point of this thesis is the determination of maximal subgroups of PSL(3,p~fc) and we have to be involved in the analysis of subgroup structures (not only maximal ones) of the group. Finally, it should mention that our philosophy comes from the well used methods in the recent papers on the symmetric graphs.