An Application of Cyclotomic Polynomial to Factorization of Abelian Groups

A finite abelian group G is said to have the Hajós-k-property (k>1) if from any decomposition of into a direct product of its subsets, it follows that one of these subsets is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai. Using some properties of the cyclotomic polynomials, we will show that the cyclic groups of orders p and groups of type (p, q), where p and q are primes have this property. We also include a partial result about groups of type (p, q), where p and q are distinct primes and α, β are integers ≥1. K A A A G ... 2 1  G i A