Countable random $p$-groups with prescribed Ulm-invariants

Abstract:

In this paper we present a probabilistic construction of countable abelian $p$-groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian $p$-groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability $1$, produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian $p$-groups. Valuated abelian $p$-groups are natural generalizations of abelian $p$-groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian $p$-groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.