Let G be a transitive
lsubgroup of the latticeordered group A(Ω) of all orderpreserving 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 nonordered permutation groups.) The
orbits of any stabilizer subgroup G_{α},α ∈ Ω, are convex and thus can be
totally ordered in a natural way. The usual pairing Δ ←→ Δ′ = {αgα ∈ Δg}
establishes an oantiisomorphism between the set of “positive” orbits and the
set of “negative” orbits. If Δ is an oblock (convex block) of G for which
ΔG_{α} = Δ, then Δ′ is also an oblock. If G_{α} has a greatest orbit Γ, then
{β ∈ ΩΓ′ < β < Γ} constitutes an oblock of G. A correspondence is established
between the centralizer Z_{A(Ω)}G and a certain subset of the fixed points of
G_{α}.
The main theorem states that every oprimitive group (G,Ω) which is not
o2transitive 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 G_{α} must consist of a set oisomorphic to the integers of
“long” (infinite) orbits with some fixed points interspersed; and there must be a
“period” z ∈ Z_{A(Ω)}G (Ω the Dedekind completion of Ω) analogous to the map βz =
β + 1 in the example. Periodic groups are shown to be lsimple, and more examples of
them are constructed.
