# Examples of uncountable abelian $p$-groups

Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable?

By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally projective groups. (The class of totally projective $p$-groups is the smallest class of abelian groups which contains the cyclic group of order $p$, is closed under taking direct sums and summands, and contains a group A if and only if it contains, for every ordinal $x$, both $p^{x}A$ and $A/p^{x}A$. It contains all countable abelian $p$-groups, and all groups in this class can be described by a recursive construction.)

The only example outside of these I know is the torsion subgroup of an unrestricted Cartesian product of cylic groups. Does anyone know of any others?

• Community wiki? Since there might not be a definitive answer, and you seem to want a list. My own answer would be: ncatlab.org/nlab/show/Pr%C3%BCfer+group Sep 6, 2012 at 11:53
• Sorry, I missed "uncountable". Sep 6, 2012 at 13:15
• Todd, the Prufer group is also known as the quasicyclic group. I mention it in the second sentence It's also countable. Any divisible $p$-group is a direct sum of copies of this group. As any divisible subgroup of an abelian group is a direct summand, my question is only really about reduced groups, those with trivial maximal divisible subgroup. I don't want to make this community-wiki yet. I'm sort of looking for a list, but I am not sure if any examples other than the ones I mention are known. Sep 6, 2012 at 13:18
• For fixed prime p, the torsion subgroup of $\prod\mathbf{Z}_{p^n}$ is an uncountable abelian p-group all of whose Ulm invariants are equal to 1; it is not isomorphic to a direct sum of cyclic groups (if it were, then the Ulm invariants would guarantee that it equals the direct sum of the $\mathbf{Z}_{p^n}$, which it plainly does not). It is also not a direct sum of indecomposable gruops. (Exercise 10.37 in Rotman's "Introduction to the Theory of Groups" 4th edition, though the text before the exercises incorrectly directs the reader to exercise 10.39). Sep 6, 2012 at 15:10
• @Arturo Magidin: I assume this group is what OP meant in the penultimate sentence of the question.
– user9072
Sep 6, 2012 at 15:12