Open Journal of Discrete Mathematics, 2011, 1, 136138 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 Email: 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 Hajóskproperty (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. K AAAG ... 21 Gi A 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 12 G G ,,mm e , r m , ,, G 12 ,,, r mm m k AA are subsets of G such that each ele ment of G can be expressed in a unique way as = 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 12 aa a A aA G G =,AGA G i to denote the number of elements of i . Similarly will donate the order of the element of G. A subset of G is called periodic if there is a nonidentity element in such that G= AA. The topic of factorizations of abelian groups arose when Hajós [2] solved a conjecture by H. Minkowski [3] 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 =, k GAAA G ,,, r eaa a G , then at least one of these subsets is a subgroup of . L. Rédei [4] 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 =, k GAAA G G A. Sands [5] classified groups with Hajós2proprty which we list below: 22 2 32 2 2 22 ,, ,, ,,,, ,,,2,2,2, ,2,2, ,2,2 ,2,2 ,,3,3 , 3,3 ,2 2,2 ,, qp qp qrpqr pp p p pp ,, 2,2, ,2 s p 2,2 , p pq 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. G Let us write. Similarly, for a subset 1 0 n i i gG ,, r g 12 g =,Ag of G, we write 0 = r i i g 2 r i g . Then we can write 12 11 1 =... , ii rr k ii AA Agg 12 i Where multiplication is carried out in the group ring . Thus, when multiplication is carried out we re gard )(GZ i as polynomials in g provided that addition of *Mathematics Subject Classification: 20K01
K. AMIN 137 the indices is carried out module n. i.e. polynomials are multiplied . 1 n mod g )...()( 21 xAxA 1x Now, if we replace g by x and write then from the relation , then we get: i i r i ixxA 1 )( 12 =, k GAAA )1mod()()( n kxxAxG As is a factor of, it follows that each irreducible divisor of 1n x 1 n x 1 nx 1 x will divide one of the polynomials i x. These irreducible poly nomials are the cyclotomic polynomials whose roots are the dth primitive roots of unity where dn and. >1d At some stage in this work, we shall need the follow ing facts about the cyclotomic polynomials. 1) The nth cyclotomic polynomial is usually denoted by n and is given by: =1 d n d n n dn xx 2) The s have integer coefficients i.e. n () xand they are irreducible and relatively prime. Slightly modifying the notation of De Bruijn [1], we also define for a divisor d of n, the polynomial 21 , 1 ==1 1 nn nd dd nd n d x xxx x n d x 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 =, k GAA A i is nonempty, there exists an ele ment ai is Ai, 1 Multiplying G by we get that Gg which is clearly normalized. .ik 11 ==Ga 12 =, k gaa 1 , , kk aA , 1 a 11 22 AaA Theorem 3.1 Let p be a prime. If G is a cyclic group of order =np , then G has the Hajoskproprty, 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: =, k GAAA 12 1 n k GxAxAxAxmodx. Now, n divides some() i x. But 1 1 == 1 p npp x xx . Thus, 1 1 1 p p x divides i x 1 . It follows that is periodic with period i A p g □ As an illustration, consider a cyclic group =Gg of order Let 3 8=2. 45 =,,, egg g and 2 =,Beg 31 . Then it is easy to verify that AB = G is a factorization of G and that A is periodic with period42 = g. We shall use the following theorem by De Bruijn [1] in showing our next result. If =npq ,1 , where are distinct primes and ,pq , xZx and n divides x, then , nq , =np xgx xhxx for some po lynomials , x hx Zx. Theorem 3.2 If G is of type , where p and q are distinct primes, then G has the Hajoskproperty, for all k, 2 ,pq 2 1< 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 [5]. Thus, we only need detail the case Say 12 =, k GAAA k G=4k =3k =G =2. A in which both factors contain elements. Again by our previous discussion, we obtain the relation pq 12 ()() ()mod(1), n GxAxAxx where It follows that 22 =npq. n divides 12 () xA x . Since 1 and 2 contain the same number of elements, we may and shall assume that n divides 1 x. Then, by De Bruijn’s result above, we get that 1, =. np nq , xfxxgxx Therefore, 1 11 = 11 nn nn pq xx Ax fxgx x . Now: 11= =1(1) pq pfqg Therefore, either 1= q and, or (1) = 0g 1=f and 0(1) = p. In the first case, , and =0gx 1 1 n np x x divides 1 x in which case 1 is periodic with period np . In the second case, and =0fx 1 1 n nq x x divides 1 x and so 1 is periodic with pe riod np .□ 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 Hajóskproperty, where =2k . 4. References [1] N. G. De Bruijn, “On the Factorization of Finite Cyclic Groups,” Indagationes Mathematicae, Vol. 15, No.4, 1953, pp. 370377. [2] G. Hajos, “Uber Einfache und Mehrfaache Bedekung des nDimensionales Raumes Mit Einem Wurfelgitter,” Ma thematics Zeitschrift, Vol. 47, No. 1, 1942, pp. 427467. doi:10.1007/BF01180974 [3] H. Minkowski, “Diophantische Approximationen,” Teu ner, Leipzig, 1907. [4] L. Redei, “Ein Beitrag Zum Problem Der Faktorisation Von Endlichen Abelschen Gruppen,” Acta Mathematics Hungarica, Vol. 1, No. 24, 1950, pp. 197207. doi:10.1007/BF02021312 [5] A. Sands, “Factorization of Finite Abelian Groups,” Acta Mathematics Hungarica, Vol. 13, No. 12, 1962, pp. 153 169. doi:10.1007/BF02033634
