Vol. 49, No. 2, 1973

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o-primitive ordered permutation groups. II

Stephen H. McCleary

Vol. 49 (1973), No. 2, 431–443

This paper is a sequal to o-primitive ordered permutation groups [Pacific J. Math., 40 (1972), S49-372]. There it was shown that if A(Ω) is the lattice-ordered group of all o-permutations of a chain Ω, and if G is an l-subgroup of A(Ω) which is periodically o-primitive (transitive and lacking proper convex blocks, but neither o-2-transitive nor regular), then the (convex) orbits of any stabilizer subgroup GαΩ, themselves form a chain o-isomorphic to the integers. Let Δ be any nonsingleton orbit of Gα. Here it is shown that Gα is faithful on Δ and that Gα|Δ is o-2-transitive and contains an element 1 of bounded support. From this it follows that all o-primitive groups (except for certain pathological o-2-transitive groups) are complete l-subgroups of A(Ω), and hence are completely distributive. When G is “full”, Gα|Δ satisfies an important “splice” property, and Gα and G are laterally complete. There is a detailed description of the unique full group G for which Δ is an α-set, and a lisling of the other “nice” permutation group representations of G.

Mathematical Subject Classification
Primary: 06A55
Received: 19 July 1973
Published: 1 December 1973
Stephen H. McCleary