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Factoring by Subsets of Cardinality Prime or Four

✍ Scribed by K. Corradi; S. Szabo

Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
362 KB
Volume
164
Category
Article
ISSN
0021-8693

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✦ Synopsis

RΓ©dei's theorem asserts that if a finite abelian group is a direct product of subsets of prime cardinality, then at least one of the factors is periodic. A theorem of A. D. Sands and S. Szabo states that if a finite elementary 2-group is factored into subsets of cardinality four, then at least one of the factors is periodic. As a common generalization of these results we prove that if a finite abelian group whose 2-component is elementary is factored into subsets whose cardinalities are of prime or four, then at least one of the factors must be periodic. i) 1994 Academic Press. Inc.

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For each positive integer j , let β j (n) := p|n p j . Given a fixed positive integer k, we show that there are infinitely many positive integers n having at least two distinct prime factors and such that β j (n) | n for each j ∈ {1, 2, . . . , k}.