The purpose of this paper is to show that the automorphism groups of many of the ‘cycle-free’ partial orders studied in [Warren, R.: The structure of k-CS-transitive cycle-free partial orders. Memoirs of the American Mathematical Society (1997), to appear] and [Creed, P., Truss, J. K. and Warren, R.: The structure of k-CS-transitive cycle-free partial orders having infinite chains, to appear] are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown in [Droste, M., Holland, W.C. and Macpherson, H.D.: Automorphism groups of infinite semilinear orders (I) and (II). Proc. London Math. Soc. 58 (1989), 454–478 and 479–494] that the automorphism group of any sufficiently transitive tree has at least normal subgroups. All the infinite chain cycle-free partial orders studied in [Creed, P., Truss, J. K. and Warren, R.: The structure of k-CS-transitive cycle-free partial orders having infinite chains, to appear] have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (≤ 2) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.
We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.
We discuss the solubility of equations of the form w = g, where w is a word (an element of a free group FX) and g is an element of a given group G. A word for which this equation is soluble for every g ∈ G is said to be universal for G. It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [Randall Dougherty and Jan Mycielski. Representations of infinite permutations by words (II). Proc. Amer. Math. Soc.127 (1999), 2233–43.], [Roger C. Lyndon. Words and infinite permutations. In Mots, Lang. Raison Calc. (Hermès, 1990), pp. 143–152.], [Jan Mycielski. Representations of infinite permutations by words. Proc. Amer. Math. Soc. 100 (1987), 237–241.], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic’ automorphisms, and elements which suitably approximate them, called ‘special’.