ON GROUP FACTORIZATIONS USING FREE MAPPINGS

We say that a collection of subsets α = [B₁,…, B_k] of a group G is a factorization if G = B₁,…,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 ℤ_p × ℤ_p and ℤ_p × ℤ_q 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.