Ulm's theorem for partially ordered structures related to simply presented abelian groups
Author:
Laurel A. Rogers
Journal:
Trans. Amer. Math. Soc. 227 (1977), 333343
MSC:
Primary 20K99; Secondary 06A75
MathSciNet review:
0442115
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Abstract: If we have an abelian pgroup G, a multiplication by p for each element of G is defined by setting , 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 pbasic tree. A natural partial order can be defined, the graph of which is a tree with 0 as root. A pbasic tree generates a simply presented abelian pgroup, and provides a natural direct sum decomposition for it. Ulm invariants may be defined directly for a pbasic 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 pbasic 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 pbasic 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/S0002994719770442115X
PII:
S 00029947(1977)0442115X
Article copyright:
© Copyright 1977
American Mathematical Society
