It is shown that if in the factorization of a finite cyclic group each factor has either prime power order or order equal to the product of two primes then one factor must be periodic. This is shown to be the best possible result of this type.  2004 Elsevier Inc. All rights reserved.

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

We will prove that two results on factoring finite abelian groups into a product of subsets, related to Hajós’s and Rédei’s theorems, can be extended for certain infinite torsion abelian groups. Mathematics Subject Classification (2000): Primary 20K01; Secondary

This note introduces Fourier transforms over finite Abelian groups, and shows how this can be used to find the period of any efficiently computable periodic function. This in particular implies an efficient quantum algorithm for factoring. In the appendix we show how this generalizes to solving the hidden subgroup problem in any Abelian group. Efficient quantum algorithms for discrete log (and ...

In this note, we define a cryptosystem based on non-commutative properties of groups. The cryptosystem is based on the hardness of the problem of factoring over these groups. This problem, interestingly, boils down to discrete logarithm problem on some Abelian groups. Further, we illustrate this method in three different non-Abelian groups GLn(Fq), UTn(Fq) and the Braid Groups.

We shall consider three results on factoring finite abelian groups by subsets. These are the Hajós’, Rédei’s and simulation theorems. As L. Fuchs has done in the case of Hajós’ theorem we shall obtain families of infinite abelian groups to which these results cannot be extended. We shall then describe classes of infinite abelian groups for which the extension does hold. MSC 2000: 20K99 (primary...

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).

Let p a be prime number. Using algebraic methods from the factorization theory of abelian groups we will prove a result about the structure of the 1-error correcting t-shift integer codes over the alphabet Zp in the special case when t is a prime. The algorithms to construct such codes can take advantage of this extra structural information in a straightforward manner and the search for these c...