Normal Subgroups of B _u Aut(Ω)

Abstract

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B _u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B _u Aut(Ω) has $2^{2^{\aleph_0}}$ normal subgroups, and there is a certain ideal ${\cal Z}$ of the lattice of subsets of ($\mathbb{Z}$), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to ${\cal Z}$.