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.

**[1]**Peter Crawley and Alfred W. Hales,*The structure of abelian 𝑝-groups given by certain presentations.*, J. Algebra**12**(1969), 10–23. MR**0238947**, https://doi.org/10.1016/0021-8693(69)90014-3**[2]**Peter Crawley and Alfred W. Hales,*The structure of abelian 𝑝-groups given by certain presentations. II.*, J. Algebra**18**(1971), 264–268. MR**0276329**, https://doi.org/10.1016/0021-8693(71)90059-7**[3]**László Fuchs,*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673****[4]**P. Hill,*On the classification of Abelian groups*, (to appear).**[5]**-,*Ulm's theorem for totally projective groups*, Notices Amer. Math. Soc.**14**(1967), 940. Abstract #652-15.**[6]**George Kolettis Jr.,*Direct sums of countable groups*, Duke Math. J**27**(1960), 111–125. MR**0110748****[7]**Helmut Ulm,*Zur Theorie der abzählbar-unendlichen Abelschen Gruppen*, Math. Ann.**107**(1933), no. 1, 774–803 (German). MR**1512826**, https://doi.org/10.1007/BF01448919**[8]**Elbert A. Walker,*The groups ᵦ*, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 245–255. MR**0364497**

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

Article copyright:
© Copyright 1977
American Mathematical Society