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The Generalized Kulikov Criterion

Published online by Cambridge University Press:  20 November 2018

Charles Megibben
Charles Megibben*
Affiliation:
Vanderbilt University, Nashville, Tennessee
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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<ωSn 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<λSα where, for each α, Sα – {0} ⊆ pαG – pα+lG. 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1192 - 1205
Copyright
Copyright © Canadian Mathematical Society 1969

References

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