The Stabilizer Subgroups And The Congruences Of The L-permutation Group

W.C.Holland[1963] has proven that every lattice-ordered group is (isomorphic to) a subgroup of the lattice-ordered group of all order-preserving permutations of a chain. So lattice-ordered permutation groups are important tools for describing the structure of lattice-ordered groups. In this paper, what we discuss is on the lattice-ordered permutation groups.In chapter two of this paper, we introduce the transitivity, which is an important definition in the 1-permutation group. And then we give some necessary and sufficient conditions for the transitive 1-permutation groups to be 2-transitive by using the stabilizer subgroups of the transitive 1-permutation groups. We also give some necessary and sufficient conditions for the transitive 1-permutation groups to be normal-valued.In chapter three of this paper, we introduce the convex congruences of 1-permutation groups. Then we research the classifications and the congruences of the transitive 1-permutation group, which is determined by the block. Furthermore, we discuss the property of the congruences and the blocks, and relationships between them.