The fully invariant subgroups of local Warfield groups

Files, Steve

doi:10.1090/S0002-9939-97-03999-3

The fully invariant subgroups of local Warfield groups
HTML articles powered by AMS MathViewer

by Steve T. Files PDF

Proc. Amer. Math. Soc. 125 (1997), 3515-3518 Request permission

Abstract:

We prove that every fully invariant subgroup of a $p$-local Warfield abelian group is the direct sum of a Warfield group and an $S$-group. This solves a problem posed some time ago by R. B. Warfield, and finalizes recent work of M. Lane concerning the fully invariant subgroups of balanced projective groups.

References

S. Files, On transitive mixed abelian groups, pp. 243–251 in Abelian Groups and Modules: Proceedings of the 1995 Colorado Springs Conference, Marcel Dekker, New York, 1996.
László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
László Fuchs, Abelian $p$-groups and mixed groups, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 70, Presses de l’Université de Montréal, Montreal, Que., 1980. MR 569744
Paul Hill and Charles Megibben, Axiom $3$ modules, Trans. Amer. Math. Soc. 295 (1986), no. 2, 715–734. MR 833705, DOI 10.1090/S0002-9947-1986-0833705-6
Roger Hunter, Fred Richman, and Elbert Walker, Warfield modules, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 87–123. MR 0506216
Roger Hunter, Fred Richman, and Elbert Walker, Existence theorems for Warfield groups, Trans. Amer. Math. Soc. 235 (1978), 345–362. MR 473044, DOI 10.1090/S0002-9947-1978-0473044-4
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
Mark Lane, Isotype submodules of $p$-local balanced projective groups, Trans. Amer. Math. Soc. 301 (1987), no. 1, 313–325. MR 879576, DOI 10.1090/S0002-9947-1987-0879576-4
Mark Lane, Fully invariant submodules of $p$-local balanced projective groups, Rocky Mountain J. Math. 18 (1988), no. 4, 833–841. MR 1002179, DOI 10.1216/RMJ-1988-18-4-833
Robert O. Stanton, Relative $S$-invariants, Proc. Amer. Math. Soc. 65 (1977), no. 2, 221–224. MR 447206, DOI 10.1090/S0002-9939-1977-0447206-0
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 291284, DOI 10.1090/S0002-9904-1972-12870-2
R. B. Warfield Jr., Classification theory of abelian groups. I. Balanced projectives, Trans. Amer. Math. Soc. 222 (1976), 33–63. MR 422455, DOI 10.1090/S0002-9947-1976-0422455-X
Robert B. Warfield Jr., The structure of mixed abelian groups, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 1–38. MR 0507073