The Factorization of Abelian Groups

If G is a nite abelian group and n > 1 is an integer, we say that G is n-good, if from each factorizationG = A1A2 An of G into direct product of subsets, it follows that at least one of the subsets Ai is periodic, in the sense that there exists x 2 G feg such that xAi = Ai. In this paper, we shall study some 3-groups with respect to this property. Acta Mathematica Academiae Paedagogicae Ny regyh aziensis 15 (1999), 9{18 www.bgytf.hu/~amapn 0. Notations and definitions Throughout this paper, G will denote a nite abelian group, e its identity element, and if a is an element of G, then jaj denotes its order. Furthermore, for a subset A of G, jAj will denote the number of the elements in A. G is said to be of type (p 1 1 ; p 2 2 ; p 3 3 ; : : : ; p S 1 ) if it is the direct product of cyclic groups of orders p 1 1 ; p 2 2 ; p 3 3 ; : : : ; p S 1 , where pi are primes. G = A1 An is said to be a factorization of G if every element a of G has a unique representation of the form a = a1 an, where ai 2 Ai. If in addition each Ai also contains e, then the factorization G = A1 An is said to be a normalized factorization. A subset A of G is said to be periodic if there is a non-identity element x in G such that xA = A. Such an element x when it exists is called a period for A. A subset A of G of the form A = fe; a; a; : : : ; ag is called cyclic; here k is an integer with k < jaj. A subset A of G is called simulated if A = fe; a; a; : : : ; a dg. We observe that if d = e, then A = < a > the subgroup generated by a. Otherwise, A di ers from the subgroup < a > generated by a in the element a d. The subgroup < a > is referred to as a corresponding subgroup of A. If A and A are subsets of G such that for every subset B of G, whenever AB = G is a factorization of G, then so is AB, then we say that A is replaceable by A. A group G is said to be n-good if from each factorization G = A1 An it follows that at least one of the Ai is periodic. Otherwise G is said to be n-bad. Furthermore, we will say G is totally-good if it is n-good for all possible values of n. 1991 Mathematics Subject Classi cation. 20K01, 05E99.