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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Ulm’s theorem for partially ordered structures related to simply presented abelian $p$-groups
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by Laurel A. Rogers PDF
Trans. Amer. Math. Soc. 227 (1977), 333-343 Request permission

Abstract:

If we have an abelian p-group G, a multiplication by p for each element of G is defined by setting $px = x + x + \cdots + x$, where p is the number of terms in the sum. If we forget about the addition on G, and just keep the multiplication by p, we have the algebraic structure called a p-basic tree. A natural partial order can be defined, the graph of which is a tree with 0 as root. A p-basic tree generates a simply presented abelian p-group, and provides a natural direct sum decomposition for it. Ulm invariants may be defined directly for a p-basic tree so that they are equal to the Ulm invariants of the corresponding group. A central notion is that of a stripping function between two p-basic trees. Given a stripping function from X onto Y we can construct an isomorphism between the groups corresponding to X and Y; in particular, X and Y have the same Ulm invariants. Conversely, if X and Y have the same Ulm invariants, then there is a map from X onto Y that is the composition of two stripping functions and two inverses of stripping functions. These results constitute Ulm’s theorem for p-basic trees, and provide a new proof of Ulm’s theorem for simply presented groups.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 227 (1977), 333-343
  • MSC: Primary 20K99; Secondary 06A75
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0442115-X
  • MathSciNet review: 0442115