A classification theorem for abelian 𝑝-groups

Warfield, R. B.

doi:10.1090/S0002-9947-1975-0372071-2

A classification theorem for abelian $p$-groups
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Trans. Amer. Math. Soc. 210 (1975), 149-168 Request permission

Abstract:

A new class of Abelian $p$-groups, called $S$-groups, is studied, and the groups in this class are classified in terms of cardinal invariants. The class of $S$-groups includes Nunke’s totally projective $p$-groups. The invariants consist of the Ulm invariants (which Hill has shown can be used to classify the totally projective groups) together with a new sequence of invariants indexed by limit ordinals which are not cofinal with $\omega$. The paper includes a fairly complete discussion of dense isotype subgroups of totally projective $p$-groups, including necessary and sufficient conditions for two of them to be congruent under the action of an automorphism of the group. It also includes an extension of Ulm’s theorem to a class of mixed modules over a discrete valuation ring.

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Classification theorems for infinite Abelian groups