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Finite Abelian groups and factorization problems. II. (English) Zbl 1164.20358
Introduction: In part I [W. Narkiewicz, ibid. 42, 319-330 (1979; Zbl 0514.12004)], several combinatorial constants associated with finite Abelian groups were defined. All of them were connected with factorization properties in algebraic number fields, arising as exponents of \(\log x\) and \(\log\log x\) in various asymptotic formulas. We pursue now this topic and consider the constant \(a_1(A)\) which was defined as the maximal length of a complex with a strongly unique factorization in a finite Abelian group \(A\). We obtain a simpler equivalent definition of it, improve the upper bound obtained in [loc. cit.], and compute the exact value for it in certain cases.

MSC:
20K01 Finite abelian groups
11R27 Units and factorization
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
11R11 Quadratic extensions
11N45 Asymptotic results on counting functions for algebraic and topological structures
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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