An Application of Cyclotomic Polynomial to Factorization of Abelian Groups

Abstract

If a finite abelian group <i>G</i> is a direct product of its subsets such that <i>G</i> = <i>A</i>₁···<i>A</i>_i···<i>A</i>_n, <i>G</i> is said to have the Hajos-<i>n</i>-proprty if it follows that one of these subsets, say <i>A_i</i> is periodic, meaning that there exists a nonidentity element <i>g</i> in <i>G</i> such that <i>gA_i</i> = <i>A_i</i> . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders <i>p^α</i> and groups of type (<i>p²</i>,<i>q²</i>) and (<i>p^α</i>,<i>p^β</i>) where <i>p</i> and <i>q</i> are distinct primes and <i>α</i>, <i>β</i> integers ≥ 1 have this property.