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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ulm's theorem for partially ordered structures related to simply presented abelian $ p$-groups


Author: Laurel A. Rogers
Journal: Trans. Amer. Math. Soc. 227 (1977), 333-343
MSC: Primary 20K99; Secondary 06A75
MathSciNet review: 0442115
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0442115-X
PII: S 0002-9947(1977)0442115-X
Article copyright: © Copyright 1977 American Mathematical Society