Communicated by I. M. IsaacsWe say that a collection of subsets α = [B 1 , . . . , B k ] of a group G is a factorization if G = B 1 , . . . , B k and each element of G is expressed in a unique way in this product. By using a special type of mappings between groups A and B, called free mappings, we exhibit an algorithmic way to construct nontrivial factorizations of a group G, such that G ∼ = A × B. In Lemma 3.2 we give a simple way to construct free mappings. It turns out that this approach has greater importance when G is an abelian group. We give illustrative examples of this method in the cases Zp × Zp and Zp × Zq where p and q are different prime numbers. An interesting connection between free mappings and Rédei's theorem, with a number theoretic implication, is given. Keywords: Group factorizations; free mappings; factoring finite groups by subsets; transformations of factorizations; factorizations of direct products. 647 J. Algebra Appl. 2008.07:647-662. Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only. 648 V. Božović & N. Paceare considered more generally as sets, rather than subgroups, for example [4,5].On the other hand, much work has been done when the blocks are subgroups, see for instance [3].In the abelian case, another term for factorization is tiling. This evokes the connection to combinatorics and geometry. Indeed, about 1900, Minkowski conjectured that:Every lattice of a tiling of R n by unit cubes contains two cubes that meet in an n − 1 dimensional face.In 1938, in his PhD thesis, Hajós reformulated Minkowski's conjecture in terms of finite abelian groups. That was the beginning of the theory of factorization of abelian groups in the sense it exists now. The fact that every abelian group is isomorphic to a factor group of an integral lattice with respect to an integral sublattice, connects the vast field of tilings and abelian groups. In general, factorization questions are relevant to the theory of numbers, tilings, packings and covering problems.On the other hand, "group factorizations" is a topic that, besides its theoretical beauty, has practical use in graph theory, coding theory, number theory and modern cryptography. Group factorizations are the main tool for cryptosystems such as PGM and MST1. Therefore, finding new ways of factorization is both of great theoretical and practical interest.In our paper, we obtain factorizations of groups of the form A × B, where A and B are groups. Our approach relies on the construction of a pair of mappings between A and B given in Lemma 3.2. Although there are no restrictions on the groups A and B, it turns out that there is a greater significance of this approach in the case where A and B are abelian groups. In Sec. 2, we give an overview of the existing results that are important for our work. In Sec. 3, the concept of free mappings is introduced and a basic tool is given for constructing new factorizations. Section 4 is treating the abelian case, with particular emphasis on the illustrative...
