Narkiewicz, W.; Śliwa, J. Finite Abelian groups and factorization problems. II. (English) Zbl 1164.20358 Colloq. Math. 46, 115-122 (1982). 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. Cited in 3 ReviewsCited in 10 Documents 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 Keywords:rings of integers of algebraic number fields; asymptotic formula; unique factorizations into irreducibles; combinatorial invariants of finite Abelian groups Citations:Zbl 0514.12004 PDFBibTeX XMLCite \textit{W. Narkiewicz} and \textit{J. Śliwa}, Colloq. Math. 46, 115--122 (1982; Zbl 1164.20358) Full Text: DOI