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

  • [1] P. Crawley and A. W. Hales, The structure of abelian p-groups given by certain presentations, J. Algebra 12 (1969) 10-23. MR 39 #307. MR 0238947 (39:307)
  • [2] -, The structure of abelian p-groups given by certain presentations. II, J. Algebra 18 (1971), 264-268. MR 43 #2076. MR 0276329 (43:2076)
  • [3] L. Fuchs, Infinite abelian groups, Academic Press, New York, 1970. MR 41 #333. MR 0255673 (41:333)
  • [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] G. Kolettis, Jr., Direct sums of countable groups, Duke Math. J. 27 (1960); 111-125. MR 22 #1616. MR 0110748 (22:1616)
  • [7] H. Ulm, Zur Theorie der Abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), 774-803. MR 1512826
  • [8] E. A. Walker, The groups $ {P_\beta }$, Symposia Mathematica, Vol. 13 (Convegno di Gruppi Abeliani, INDAM, Rome, 1972), Academic Press, London, 1974, 245-255. MR 51 #751. MR 0364497 (51:751)

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

DOI: https://doi.org/10.1090/S0002-9947-1977-0442115-X
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society