From: Paul Robert BrownDate: Fri, 25 Oct 1996 10:36:36 -0700 (PDT) Subject: Re: assigning orders to permutations Hello, Forum and Derek. When 1/l + 1/m + 1/n is less than or equal to one, this can be accomplished by taking quotients of triangle groups. Let T(l,m,n) be the triangle group which acts by reflections on the Euclidean or hyperbolic plane with fundamental domain a triangle with angles pi/l, pi/m, and pi/n. T(l,m,n) contains normal surface subgroups of arbitrarily large index, so the groups we desire are automorphism groups of suitably triangulated surfaces; the finite set Omega could be taken to be some set of simplices of the surface, eg., the set of 2-cells, in which case the automorphism group acts transitively. (Note that I've chosen the surface group to be normal in the triangle group, so this is the case.) When 1/l + 1/m + 1/n > 1, the triangle group acts on a 2-sphere, so I would be interested to know what additional requirements you would like to place on the set Omega (else I'll just make Omega large by taking the disjoint union of lots of 2-spheres and letting the triangle group act diagonally). Do you want the action to be transitive on the set? Best, Paul --------------------------------------------------------