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

Author:
Laurel A. Rogers

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

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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 , 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:
https://doi.org/10.1090/S0002-9947-1977-0442115-X

Article copyright:
© Copyright 1977
American Mathematical Society