Finite Abelian groups and factorization problems. II.

*(English)*Zbl 1164.20358Introduction: 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 |