On the theory and classification of Abelian \(p\)-groups. (English) Zbl 0535.20031

Hill, Paul; Megibben, Charles

Math. Z. 190, 17-38 (1985).

We initiate a general theory of IT-groups (isotype subgroups of totally projective abelian \(p\)-groups) based on the notion of a \(c\)-valuated group and the following powerful uniqueness theorem: Isotype subgroups \(H\) and \(K\) of the totally projective group \(G\) are isomorphic under an automorphism of \(G\) if and only if \(H\) and \(K\) have the same Ulm-Kaplansky invariants and the quotients \(G/H\) and \(G/K\) are isomorphic as \(c\)-valuated groups. This theorem leads to a unification and extension of the existing classification theory of infinite abelian \(p\)-groups, yielding new characterizations of previously classified IT-groups (e.g., \(S\)-groups, \(A\)-groups) and the generalization to all IT-groups of a variety of structural properties known heretofore only for special classes of IT-groups. The underlying theme is that the theory of IT-groups is coextensive with that of \(c\)-valuated \(p\)-groups, a point of view that produces both results of a positive and a negative character; for example, an IT-group associated with a countable \(c\)-valuated group is necessarily an \(S\)-group and, on the other hand, the classification up to isomorphism of a special class of IT-groups of length \(\omega_ 1\) is equivalent in a precise sense to the classification of all abelian \(p\)-groups.

MSC:

20K10	Torsion groups, primary groups and generalized primary groups
20K99	Abelian groups
20K27	Subgroups of abelian groups

Keywords:

IT-groups; isotype subgroups of totally projective abelian \(p\)-groups; uniqueness theorem; Ulm-Kaplansky invariants; \(c\)-valuated groups

References:

[1]	Fuchs, L.: Infinite abelian groups, Vol. II. New York: Academic Press 1973 · Zbl 0257.20035
[2]	Fuchs, L.: Valued vector spaces. J. Algebra35, 23-38 (1975) · Zbl 0318.15002 · doi:10.1016/0021-8693(75)90033-2
[3]	Fuchs, L.: Extensions of isomorphisms between subgroups. Lecture Motes in Math., Vol.874, pp. 289-296. Berlin, Heidelberg, New York: Springer 1981 · Zbl 0469.20026
[4]	Griffith, P.: Infinite abelian groups. Chicago, Ill.: Univ. of Chicago Press 1970
[5]	Hill, P.: On the classification, of abelian groups. Photocopied manuscript, 1967 · Zbl 0168.27303
[6]	Hill, P.: Criteria for freeness in groups and valuated vector spaces. Lecture Notes in Math., Vol.616, pp. 140-157. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0372.20041
[7]	Hill, P.: Isotype subgroups of totally projective groups. Lecture Notes in Math., Vol.874, pp. 305-321, Berlin, Heidelberg, New York: Springer 1981 · Zbl 0466.20028
[8]	Hill, P.: The equivalence of high subgroups. Proc. Am. Math. Soc.88, 207-211 (1983) · Zbl 0522.20038 · doi:10.1090/S0002-9939-1983-0695242-4
[9]	Hill, P.: On the structure of abelianp-groups. Trans. Am. Math. Soc.288, 505-525 (1985) · Zbl 0573.20053
[10]	Hill, P., Megibben, C.: On direct sums of countable groups and generalizations. Studies on Abelian Groups, pp. 183-206. Paris: Dunod 1968 · Zbl 0203.32705
[11]	Hunter, R., Walker, E.:S-groups revisited. Proc. Am. Math. Soc.82, 13-18 (1981) · Zbl 0464.20037
[12]	Kaplansky, I., Mackey, G.: A generalization of Ulm’s Theorem. Summa Brasil. Math.2, 195-202 (1951) · Zbl 0054.01803
[13]	Kolettis, G.: Direct sums of countable groups. Duke Math. J.27, 111-125 (1960) · Zbl 0091.02802 · doi:10.1215/S0012-7094-60-02711-3
[14]	Megibben, C.: Totally Zippinp-groups. Proc. Am. Math. Soc.91, 15-18 (1984) · Zbl 0543.20036
[15]	Nunke, R.: Homology and direct sums of countable abelian groups. Math. Z.101, 182-212 (1967) · Zbl 0173.02401 · doi:10.1007/BF01135839
[16]	Richman, F., Walker, E.: Valuated groups. J. Algebra56, 145-167 (1979) · Zbl 0401.20049 · doi:10.1016/0021-8693(79)90330-2
[17]	Stanton, R.:S-groups. Preprint · Zbl 0499.05002
[18]	Ulm, H.: Zur Theorie der abzahlbar-unendlichen abelschen Gruppen., Math. Ann.107, 774-803 (1933) · JFM 59.0143.03 · doi:10.1007/BF01448919
[19]	Warfield, R.: A classification theorem for abelianp-groups. Trans. Am. Math. Soc.210, 149-168 (1975) · Zbl 0324.20058
[20]	Warfield, R.: The structure of mixed abelian groups. Lecture Notes in Math., Vol.616, pp. 1-38. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0368.20032