John R. Britnell, Mark Wildon
July 23, 2013
Abstract. For , let be the set of partitions of Ω given by the cycles of elements of G . Under the refinement order, admits join and meet operations. We say that G is join- or meet-coherent if is join- or meet-closed, respectively. The centralizer in of any permutation g is shown to be meet-coherent, and join-coherent subject to a finiteness condition. Hence if G is a centralizer in S n , then is a lattice. We prove that wreath products, acting imprimitively, inherit join-coherence from their factors. In particular automorphism groups of locally finite, spherically homogeneous trees are join-coherent. We classify primitive join-coherent groups of finite degree, and also join-coherent subgroups of S n normalizing an n -cycle. We show that if is a chain, then there is a prime p such that G acts regularly on each of its orbits as a subgroup of the Prüfer p -group, with G being isomorphic to an inverse limit of these subgroups.