Hajós-Property for Direct Product of Groups

We study decomposition of finite Abelian groups into subsets and show by examples a negative answer to the question of whether Hajós-property is inherited by direct product of groups which have Hajós-property.


Introduction
The general setting is as follows: Suppose we decompose a group G into direct product of subsets 1 2 , , , n A A A  of G in such a way that each element g in G has a unique representation of the form .The answer is rather difficult even if we do not impose many restrictions either on G or on the subsets.The most important special case has some connection with a group-theoretial formulation by G. Hajós [1] of a conjecture by H. Minkowski [2]; this is when G is a finite Abelian group and each of the subsets is of the form where k g < is an integer; here e denotes the identity element of g and g denotes order of the element g of G. Then a result due to Hajos states that one of the subsets i A must be a subgroup of G. L. Rédei [3] generalizes this result to the case when the condition on the subsets i A is that they contain a prime number of elements.
Another interesting question has also been asked by Hajos.It is concerned with the case in which G is an Abelian group and 2 n = ; the question then asked is as follows: Suppose G has a decomposition as

Does it follow that one of the subsets 1
A or 2 A is a direct product of another subset and a proper subgroup of G?
The concept of Hajós factorization begin group-theoretical but now finds applications in diverse fields such as number theory, [4] coding theory [5] and even in music [6].

Preliminaries
Throughout this paper, G will denote a finite Abelian group, e the identity of G, and if g G ∈ , then g will denote its order.We will also use A to denote the number of elements of  we say that we have a factorization of G.If in addition, each of the subsets i A contains e, we say that we have , such that gA A = .Such an element g G ∈ if it exists is called a period for A. A group G is said to be of type ( ) α ′ are non-negative integers).

Remarks
. Thus, we may assume that all factorization we consider are normalized.2) In the literature, a group G is said to be "good" if from each factorization G AB = , it follows that one the subsets A or B is periodic.
We extend the above definition as follows.

Definition
A group G has the Hajos-n-property or n-good if from any factorization is periodic.Otherwise it is n-bad.We will also say G is totallygood if it is n-good for all possible values of n.
The following results are known and will be used in this paper.

Lemma 2 [8]
A cyclic group G of order p α , where 3 p > is prime is totally-good.Lemma 3 [8] If G is of type ( ) If H is a proper subgroup of G, then there exists a non-periodic set N such that G HN = is a factorization of G, except when H is a subgroup of index 2 in an elementary abelian 2-group.

Lemma 5 [7]
If A and B are non-periodic subsets o a group G and A is contained in a subgroup H of G such that G HB = is a factorization of G, then AB is also non-periodic.

Theorem 6
If G is of type ( ) question then asked is what we can say about the subsets 1 2 , , , n A A A  A =  be a factorization of G. Now, the possible values for n are 1, 2, 3 and 4. The case 1 n = is trivial.The case 2 n = follows from Lemma 1.