Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20K99, 06A75

Retrieve articles in all journals with MSC: 20K99, 06A75

Additional Information

Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society