The Generalized Kulikov Criterion

Megibben, Charles

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1192-1205

Résumé

In 1941, Kulikov (5) showed that a p-primary abelian group G is a direct sum of cyclic groups if and only if G is the union of an ascending sequence of subgroups each of which has a finite bound on the heights of its elements. An easy reformulation of the Kulikov criterion is: A p-primary abelian group G is a direct sum of cyclic groups if and only if G[p] = ⴲn <ω Sn where, for each n, the non-zero elements of Sn have precisely height n. This statement suggests the consideration of reduced p-groups G such that G[p] = ⴲa <λ Sα where, for each α, S α – {0} ⊆ pαG – pα +l G. We shall call such p-groups summable (the term principal p-group has been used by Honda (4)). Recall that the length of a reduced p-group G is the first ordinal λ such that pλG = 0.

Détail

DOI : 10.4153/CJM-1969-132-9

Bibliographie

[1] 1. Hill, P., Isotype subgroups of direct sums of countable groups (to appear). Google Scholar

[2] 2. Hill, P., A summable C-group, Proc. Amer. Math. Soc. (to appear). Google Scholar

[3] 3. Hill, P. and Megibben, C., On direct sums of countable groups and generalizations, pp. 183–206, Etudes sur les Groupes Abéliens, Paris, 1968. Google Scholar

[4] 4. Honda, K., On the structure of abelian p-groups, pp. 81–86, Proc. Colloq. on Abelian Groups, Budapest, 1964. Google Scholar

[5] 5. Kulikov, L., Zur Théorie der abelschen Gruppen von beliebizer Machtizheit, Mat. Sb. 9 (1941), 165–181. Google Scholar

[6] 6. Megibben, C., A generalization of the classical theory of primary groups (to appear). Google Scholar

[7] 7. Nunke, R., Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212. Google Scholar

[9] 9. Nunke, R., On the structure of Tor II, Pacific J. Math. 22 (1967), 453–464 Google Scholar