 Open Journal of Discrete Mathematics, 2011, 1, 136-138 doi:10.4236/ojdm.2011.13017 Published Online October 2011 (http://www.SciRP.org/journal/ojdm) Copyright © 2011 SciRes. OJDM An Application of Cyclotomic Polynomial to Factorization of Finite Abelian Groups* Khalid Amin University of Bahrain, Department of Mathematics, Sakhir, Kingdom of Bahrain E-mail: kamin@sci.uob.bh Received June 21, 2011; revised July 25, 2011; accepted August 21, 2011 Abstract A finite abelian group Gis 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 subsetsis 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 pα and groups of type (p2, q2), where p and q are primes have this property. We also include a partial result about groups of type (pα, qβ), where p and q are distinct primes and α, β are integers ≥1. KAAAG ...21GiA Keywords: Factorization of Finite Abelian Groups, Hajós Property 1. Introduction Let be a finite abelian group with identity element . If is a direct product of cyclic groups of orders 12 , we say that is of type. If 12GG,,mme,rm,,,G12,,,rmm mkAAA are subsets of G such that each ele-ment g of G can be expressed in a unique way as =g k, where ii, we write 12 k and say that we have a factorization of . If in addition each contains the identity element , we say that we have a normalized factorization of . We will use 12aaaAaAGG=,AGAGiA to denote the number of elements of iA. Similarly g will donate the order of the element g of G. A subset A of G is called periodic if there is a non-identity element g in such that G=gAA. The topic of factorizations of abelian groups arose when Hajós  solved a conjecture by H. Minkowski  concerning lattice tiling after transforming it into a theo-rem about finite abelian groups. For reference, we state this theorm below: If is a factorization of a finite abe-lian group , where each of the subsests is of the form 12=,kGAAAG,,, reaa aG, then at least one of these subsets is a subgroup of . L. Rédei  generalized this to: If is a factorization of a finite abe-lian group , where each of the subsets has a prime number of elements and contains the identity e, then at least one of these subsets is a subgroup of . 12=,kGAAAGGA. Sands  classified groups with Hajós-2-proprty which we list below:   22 232 2222,, ,, ,,,,,,,2,2,2, ,2,2, ,2,2,2,2 ,,3,3 , 3,3 ,22,2 ,,qp qp qrpqrpp pppp,,2,2,,2sp2,2,ppq where p, q, r and s are primes and 1 is an integer. 2. Preliminaries Let be a cyclic group of order n, with generator g. GLet us write. Similarly, for a subset 10niigG,,rg12g=,Ag of G, we write 0=riiAg2rig. Then we can write 1211 1=... ,iirrkiiAA Agg 12i Where multiplication is carried out in the group ring . Thus, when multiplication is carried out we re-gard )(GZiA as polynomials in g provided that addition of *Mathematics Subject Classification: 20K01 K. AMIN 137the indices is carried out module n. i.e. polynomials are multiplied . 1nmod g)...()( 21 xAxA1xNow, if we replace g by x and write then from the relation , then we get: iiriixxA1)(12=,kGAAA )1mod()()(  nkxxAxG As is a factor of, it follows that each irreducible divisor of 1nx1nx1nx1x will divide one of the polynomials iAx. These irreducible poly-nomials are the cyclotomic polynomials whose roots are the d-th primitive roots of unity where dn and. >1dAt some stage in this work, we shall need the follow-ing facts about the cyclotomic polynomials. 1) The n-th cyclotomic polynomial is usually denoted by nx and is given by: =1dndnndnxx 2) The xs have integer coefficients i.e. nx ()Zxand they are irreducible and relatively prime. Slightly modifying the notation of De Bruijn , we also define for a divisor d of n, the polynomial 21,1==11nnndddnd ndxxxxxndx 3. Results Before we embark on showing our results, we must men-tion that all factorizations can be assumed to be normal-ized, for if is a factorization of G, then since each 12=,kGAA AiA is non-empty, there exists an ele-ment ai is Ai, 1 Multiplying G by we get that Gg which is clearly normalized. .ik11==Ga12=,kgaa1, ,kkaA,1a1122AaATheorem 3.1 Let p be a prime. If G is a cyclic group of order =np, then G has the Hajos-k-proprty, for all k, 1< k. Proof Let G be generated by g and consider the fac-torization 12 of G. Our previous discus-sion leads to the following congruence relation: =,kGAAA 12 1nkGxAxAxAxmodx. Now, nx divides some()iAx. But  11==1pnppxxxx . Thus, 111ppxxdividesiAx1. It follows that is periodic with periodiApg□ As an illustration, consider a cyclic group =Gg of order Let 38=2.45=,,,Aegg g and 2=,Beg31. Then it is easy to verify that AB = G is a factorization of G and that A is periodic with period42=gg. We shall use the following theorem by De Bruijn  in showing our next result. If =npq,1, where are distinct primes and ,pq, FxZx and nx divides Fx, then ,nq,=npFxgx xhxx for some po-lynomials ,gx hx Zx. Theorem 3.2 If G is of type, where p and q are distinct primes, then G has the Hajos-k-property, for all k, 2,pq21< 4k. Proof Consider a factorization 12 of . The case is true by Redei's theorem. The case , is true by Redei theorem and Theorem 2 of Sands . Thus, we only need detail the case Say 12=,kGAAAkG=4k=3k=G=2.AA in which both factors contain elements. Again by our previous discussion, we obtain the relation pq12()() ()mod(1),nGxAxAxx where It follows that 22=npq.nx divides 12()AxA x . Since 1A and 2A contain the same number of elements, we may and shall assume that nx divides1Ax. Then, by De Bruijn’s result above, we get that 1,=.np nq,Axfxxgxx Therefore,  111=11nnnnpqxxAx fxgxxx. Now: 11= =1(1)Apq pfqg Therefore, either 1=fq and, or (1) = 0g1=f and 0(1) =gp. In the first case, , and =0gx11nnpxx divides 1Ax in which case 1A is periodic with period npg. In the second case, and =0fx11nnqxx divides1Ax and so 1A is periodic with pe-riod npg.□ Corollary 3.1 If G is of type,pq, where p and q are distinct primes, and , are integers , then G 1Copyright © 2011 SciRes. OJDM K. AMIN Copyright © 2011 SciRes. OJDM 138 has the Hajós-k-property, where =2k. 4. References  N. G. De Bruijn, “On the Factorization of Finite Cyclic Groups,” Indagationes Mathematicae, Vol. 15, No.4, 1953, pp. 370-377.  G. Hajos, “Uber Einfache und Mehrfaache Bedekung des n-Dimensionales Raumes Mit Einem Wurfelgitter,” Ma-thematics Zeitschrift, Vol. 47, No. 1, 1942, pp. 427-467. doi:10.1007/BF01180974  H. Minkowski, “Diophantische Approximationen,” Teu-ner, Leipzig, 1907.  L. Redei, “Ein Beitrag Zum Problem Der Faktorisation Von Endlichen Abelschen Gruppen,” Acta Mathematics Hungarica, Vol. 1, No. 2-4, 1950, pp. 197-207. doi:10.1007/BF02021312  A. Sands, “Factorization of Finite Abelian Groups,” Acta Mathematics Hungarica, Vol. 13, No. 1-2, 1962, pp. 153- 169. doi:10.1007/BF02033634