(Prior to publication, this paper should be cited as arXiv:0712.4142.)

Let  G  be a transitive group of permutations of a finite set  X,  and suppose that some element of  G  has at most two orbits on  X.  We prove that any two maximal chains of groups between  G  and a point-stabilizer of  G  have the same length, and the same sequence of relative indices between consecutive groups (up to permutation). We also deduce the same conclusion when  G  has a transitive quasi-Hamiltonian subgroup.

Previously the first result had been proved in special cases by Zieve and Pakovich, as a consequence of results on decompositions of Laurent polynomials. Conversely, this result has the following consequence. Let  C  and  D  be smooth, projective, geometrically irreducible curves over a field  K,  and let  f : C → D  be a nonconstant separable rational map defined over  K.  Suppose that some closed point of  D  is tamely ramified in  f , and lies under at most two closed points of  C.  Let  C → A₁ → A₂ → ...→ D  and C → B₁ → B₂ → ...→ D  be maximal decompositions of  f  as the composition of rational maps of degree more than 1. Then these decompositions have the same length, and (up to permutation) the same sequence of degrees of the involved indecomposable maps.

The first result was discovered independently and simultaneously by Muzychuk and Pakovich, via a different argument.

The second result was previously proved by Ritt for groups with a transitive cyclic subgroup, and by Müller for groups with a transitive abelian subgroup. More generally, a group  I  is quasi-Hamiltonian if any two subgroups  A, B  of  I  satisfy  AB = BA.  Plainly any abelian group is quasi-Hamiltonian, but there are also non-abelian examples, for instance the direct product of the order-8 quaternion group with any abelian group containing no element of order 4.

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