A new upper bound for the cross number of finite abelian groups
- 61 Downloads
In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite abelian groups. Given a finite abelian group, this upper bound appears to depend only on the rank and the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite abelian groups holds asymptotically in at least two different directions.
KeywordsAbelian Group Cyclic Group Direct Summand Finite Abelian Group Cross Number
Unable to display preview. Download preview PDF.
- S. T. Chapman, On the Davenport constant, the cross number, and their application in factorization theory, in Zero-dimensional Commutative Rings (Knoxville, TN, 1994), Lecture Notes in Pure and Applied Mathematics, Dekker, New-York, Vol. 171, 1995, pp. 167–190.Google Scholar
- P. van Emde Boas, A combinatorial problem on finite abelian groups II, Reports ZW-1969-007, Math. Centre, Amsterdam, 1969.Google Scholar
- A. Geroldinger and F. Halter-Koch, Non-unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Vol. 278, Chapman & Hall/CRC, 2006.Google Scholar
- P. Samuel, Théorie Algébrique des Nombres, Hermann, 2003.Google Scholar
- A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1998.Google Scholar
- G. Tenenbaum, Introduction à la Théorie Analytique et Probabiliste des Nombres, Cours spécialisés, SMF, 1995.Google Scholar