By Ulm's theorem, countable reduced abelian p-groups are characterized, uniquely up to isomorphism, by their Ulm invariants. Given a sequence f of Ulm invariants, we provide a probabilistic construction of a countable abelian p-group Gf, having the set of natural numbers as its domain, with Ulm invariants ≤ f. We then show that with probability 1, Gf has precisely f as its sequence of Ulm invariants. This establishes the existence part of Ulm's theorem in a probabilistic way. We also develop new results for valuated abelian p-groups which are essential for our construction.
Joint work with Ruediger Goebel (Essen).