Existence theorems for Warfield groups

Authors:
Roger Hunter, Fred Richman and Elbert Walker

Journal:
Trans. Amer. Math. Soc. **235** (1978), 345-362

MSC:
Primary 20K25

MathSciNet review:
0473044

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Abstract: Warfield studied *p*-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

**[1]**László Fuchs,*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673****[2]**Roger Hunter, Fred Richman, and Elbert Walker,*Simply presented valuated abelian 𝑝-groups*, J. Algebra**49**(1977), no. 1, 125–133. MR**0507069****[3]**Fred Richman,*The constructive theory of countable abelian 𝑝-groups*, Pacific J. Math.**45**(1973), 621–637. MR**0344088****[4]**Fred Richman,*The constructive theory of 𝐾𝑇-modules*, Pacific J. Math.**61**(1975), no. 1, 263–274. MR**0399069****[5]**Fred Richman and Elbert A. Walker,*Ext in pre-Abelian categories*, Pacific J. Math.**71**(1977), no. 2, 521–535. MR**0444742****[6]**-,*Valuated groups*(to appear).**[7]**Laurel A. Rogers,*Ulm’s theorem for partially ordered structures related to simply presented abelian 𝑝-groups*, Trans. Amer. Math. Soc.**227**(1977), 333–343. MR**0442115**, 10.1090/S0002-9947-1977-0442115-X**[8]**Joseph Rotman and Ti Yen,*Modules over a complete discrete valuation ring*, Trans. Amer. Math. Soc.**98**(1961), 242–254. MR**0122895**, 10.1090/S0002-9947-1961-0122895-1**[9]**R. O. Stanton,*An invariant for modules over a discrete valuation ring*, Proc. Amer. Math. Soc.**49**(1975), 51–54. MR**0360572**, 10.1090/S0002-9939-1975-0360572-8**[10]**Elbert A. Walker,*Ulm’s theorem for totally projective groups*, Proc. Amer. Math. Soc.**37**(1973), 387–392. MR**0311805**, 10.1090/S0002-9939-1973-0311805-3**[11]**R. B. Warfield Jr.,*Classification theorems for 𝑝-groups and modules over a discrete valuation ring*, Bull. Amer. Math. Soc.**78**(1972), 88–92. MR**0291284**, 10.1090/S0002-9904-1972-12870-2**[12]**-,*Classification theory of abelian groups*. II:*Local theory*(to appear).

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DOI:
https://doi.org/10.1090/S0002-9947-1978-0473044-4

Article copyright:
© Copyright 1978
American Mathematical Society