The recovery of some abelian groups from their socles

Hill, Paul

doi:10.1090/S0002-9939-1982-0674080-1

The recovery of some abelian groups from their socles
HTML articles powered by AMS MathViewer

by Paul Hill PDF

Proc. Amer. Math. Soc. 86 (1982), 553-560 Request permission

Abstract:

The first main result of this paper is that summable isotype subgroups inherit total projectivity. This means: ($\ast$) if an isotype subgroup of a totally projective group has a free socle (viewed as a valuated vector space) then the socle completely determines the subgroup up to isomorphism. The next major result is that ($\ast$) does not generalize to the case where the socle is that of a totally projective group of length exceeding $\Omega$, nor does ($\ast$) generalize to the case where the socle is that of an $S$-group of length $\Omega$. Finally, it is shown that if, in addition to $H$ being isotype in a d.s.c. group $G$, it is also known that $K/H$ is divisible where $K$ is the closure of $H$ in $G$ relative to the ${p^\Omega }$-topology, then ($\ast$) again prevails when the socle is that of an $S$-group.

References

Doyle O. Cutler, Another summable $C_{\Omega }$-group, Proc. Amer. Math. Soc. 26 (1970), 43–44. MR 262355, DOI 10.1090/S0002-9939-1970-0262355-1
László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
László Fuchs, Subfree valued vector spaces, 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. 158–167. MR 0480700

On the classification of abelian groups

Paul Hill, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), 281–290. MR 240198
Paul Hill, A summable $C_{\Omega }$-group, Proc. Amer. Math. Soc. 23 (1969), 428–430. MR 245674, DOI 10.1090/S0002-9939-1969-0245674-6

Criteria for freeness in abelian groups and valuated vector spaces

Paul Hill, Isotype subgroups of totally projective groups, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 305–321. MR 645937
Paul Hill and Charles Megibben, On direct sums of countable groups and generalizations, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 183–206. MR 0242943
Roger Hunter and Elbert Walker, $S$-groups revisited, Proc. Amer. Math. Soc. 82 (1981), no. 1, 13–18. MR 603592, DOI 10.1090/S0002-9939-1981-0603592-0
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
Irving Kaplansky and George W. Mackey, A generalization of Ulm’s theorem, Summa Brasil. Math. 2 (1951), 195–202. MR 49165
George Kolettis Jr., Direct sums of countable groups, Duke Math. J. 27 (1960), 111–125. MR 110748
R. J. Nunke, Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. MR 218452, DOI 10.1007/BF01135839
Fred Richman and Elbert A. Walker, Valuated groups, J. Algebra 56 (1979), no. 1, 145–167. MR 527162, DOI 10.1016/0021-8693(79)90330-2
Helmut Ulm, Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), no. 1, 774–803 (German). MR 1512826, DOI 10.1007/BF01448919
R. B. Warfield Jr., A classification theorem for abelian $p$-groups, Trans. Amer. Math. Soc. 210 (1975), 149–168. MR 372071, DOI 10.1090/S0002-9947-1975-0372071-2