Báo cáo toán học: Fibonacci Length of Direct Products of Groups

Đối với một tổ chức phi-abelian hữu hạn nhóm G = a1, a2,... , Chiều dài Fibonacci của G với sự tôn trọng để tạo ra lệnh thiết lập A = {a1, a2,... , Một} là l số nguyên như vậy trình tự của các yếu tố xi = ai, 1 ≤ i ≤ n, xn + i = nj = 1 xi + k-1, i ≥ 1, G, các phương trình xl + i = ai.
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Báo cáo toán học: " Fibonacci Length of Direct Products of Groups" 9LHWQDP -RXUQDOVietnam Journal of Mathematics 33:2 (2005) 189–197 RI 0$7+(0$7,&6 ‹ 9$67 Fibonacci Length of Direct Products of Groups H. Doostie1 and M. Maghasedi2 1 Mathematics Department, Teacher Training University, 49 Mofateh Ave., Tehran 15614, Iran 2 Mathematics Section Doctoral Research Department, Islamic Azad University, P.O. Box. 14515-775, Tehran, Iran Received July 7, 2004 Revised December 4, 2004Abstract. For a non-abelian finite group G = a1 , a2 , . . . , an the Fibonacci lengthof G with respect to the ordered generating set A = {a1 , a2 , . . . , an } is the leastinteger l such that for the sequence of elements xi = ai , 1 ≤ i ≤ n, xn+i = n j =1 xi+j −1 , i ≥ 1, of G, the equations xl+i = ai , 1 ≤ i ≤ n hold. The questionposed in 2003 by P. P. Campbell that ”Is there any relationship between the lengthsof finite groups G, H and G × H ?” In this paper we answer this question when atleast one of the groups is a non-abelian 2-generated group.1. IntroductionLet G = A be a finite non-abelian group where, A = {a1 , a2 , . . . , an } is anordered generating set. The sequence n xi = ai , 1 ≤ i ≤ n, xn+i = xi+j −1 , i ≥ 1 j =1of the elements of G, denoted by FA (G), is called the Fibonacci orbit of G withrespect to A, and the least integer l for which the equations xl+i = xi , 1 ≤i ≤ n hold, is called the Fibonacci length of G with respect to A and will bedenoted by LENA (G). The notions of basic Fibonacci orbit and basic Fibonaccilength are almost similar. Indeed, the basic Fibonacci orbit of length m is alsodefined to be the same sequence of the elements of G such that m is the leastinteger where the equations x1 θ = xm+1 , x2 θ = xm+2 , . . . , xn θ = xm+n hold190 H. Doostie and M. Maghasedifor some θ ∈ Aut(G). The integer m is called the basic Fibonacci length ofG with respect to A and will be denoted by BLENA (G). It is proved in [2]that BLENA (G) divides LENA (G), for 2-generated finite groups. ObviouslyA ∪ {b1 , b2 , . . . , bk }, k ≥ 1, b1 = b2 = · · · = bk = 1 is also a generating set forG, however, the Fibonacci lengths with respect to A and with respect to thisgenerating set are different. We will use this new generating set to make possibleour calculations. Since 1990 the Fibonacci length has been studied and calculated for certainclasses of finite groups ( one may see [1, 2, 4, 7], for examples), and certainoriginal questions have been posed by Campbell in [5]. We answer one of thesequestions which is: how can one calculate the Fibonacci length of G×H (externaldirect product of groups G and H ) in terms of the Fibonacci lengths of G andH? For a finite number of the groups G1 , G2 , . . . , Gk we use the notation Drk Gifor the external direct product (or, simply the direct product) G1 × G2 ×· · ·× Gk . Our first attempt is to calculate LEN{a,b,e} (G × H ) where G = a, b is anon-abelian finite group and H = e is a cyclic group of order m; then we willcalculate LEN{a,b,c,d}(G × H ) where G = a, b and H = c, d are non-abelianfinite groups. For every positive integer m, we define the positive integer k (m, 3) to be theminimal length of the period of series (gi mod m)+∞ , where −∞ g0 = g1 = 0, g2 = 1, gi = gi−1 + gi−2 + gi−3 .This number is similar to the Wall number k (m) of Wall [10], where the Wallnumber is defined for the series (fi mod m)+∞ such that −∞ f0 = f1 = 1, fi = fi−1 + fi−2 .Following Wall [10] one may also prove the existence of k (m, 3) for every positiveinteger m. We will use 1 for 1G , the identity of the group G. Our main resultsare the following propositions. Propositions A, B and C give explicit formulasfor computing the lengths of direct products of some groups, and PropositionsD and E are the generalization of [2] for LEN{a,b} (D2n ) and LEN{a,b} (Q2n )(see [2]).Proposition A. For every 2-generated non-abelian finite group G = a, b andevery cyclic group H = e of order m, LEN{a,b,e} (G × H ) = l.c.m.(k (m, 3), LEN{a,b,1} (G)).Proposition B. For every 2-generated non-abelian finite groups G = a, b andH = c, d , LEN{a,b,c,d}(G × H ) = l.c.m.(LEN{a,b,1,1} (G), LEN{1,1,c,d}(H ...