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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A classification theorem for abelian $ p$-groups

Author: R. B. Warfield
Journal: Trans. Amer. Math. Soc. 210 (1975), 149-168
MSC: Primary 20K10
MathSciNet review: 0372071
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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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Keywords: Ulm's theorem, totally projective groups, isotype subgroups, invariants, cotorsion groups, automorphisms of Abelian $ p$-groups
Article copyright: © Copyright 1975 American Mathematical Society