Vol. 40, No. 2, 1972

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0-primitive ordered permutation groups

Stephen H. McCleary

Vol. 40 (1972), No. 2, 349–372

Let G be a transitive l-subgroup of the lattice-ordered group A(Ω) of all order-preserving permutations of a chain Q. (In fact, many of the results are generalized to partially ordered sets Ω and transitive groups G such that β < γ implies βg = γ for some positive g G, thus encompassing some results on non-ordered permutation groups.) The orbits of any stabilizer subgroup GαΩ, are convex and thus can be totally ordered in a natural way. The usual pairing Δ ←→ Δ= {αg|α Δg} establishes an o-anti-isomorphism between the set of “positive” orbits and the set of “negative” orbits. If Δ is an o-block (convex block) of G for which ΔGα = Δ, then Δis also an o-block. If Gα has a greatest orbit Γ, then {β Ω|Γ< β < Γ} constitutes an o-block of G. A correspondence is established between the centralizer ZA(Ω)G and a certain subset of the fixed points of Gα.

The main theorem states that every o-primitive group (G,Ω) which is not o-2-transitive or regular looks strikingly like the only previously known example, in which Ω is the reals and G = {f A(Ω)|(β + 1)f = βf + 1 for all β Ω}. The “configuration” of orbits of Gα must consist of a set o-isomorphic to the integers of “long” (infinite) orbits with some fixed points interspersed; and there must be a “period” z ZA(Ω)G (Ω the Dedekind completion of Ω) analogous to the map βz = β + 1 in the example. Periodic groups are shown to be l-simple, and more examples of them are constructed.

Mathematical Subject Classification
Primary: 06A55
Received: 10 February 1970
Revised: 11 March 1971
Published: 1 February 1972
Stephen H. McCleary