tailieunhanh - 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. | 33 2 2005 189-197 V í e t mi ai m J o u r mi ai l of MATHEMATICS ê VAST 2005 Fibonacci Length of Direct Products of Groups H. Doostie1 and M. Maghasedi2 Mathematics Department Teacher Training University 49 Mofateh Ave. Tehran 15614 Iran 2Mathematics Section Doctoral Research Department Islamic Azad University . Box. 14515-775 Tehran Iran Received July 7 2004 Revised December 4 2004 Abstract. For a non-abelian finite group 1 2. the Fibonacci length of with respect to the ordered generating set A 1 2. is the least integer such that for the sequence of elements . . . 1 . . 1 of the equations . . .hold. The question posed in 2 0 0 3 by P. P. Campbell that Is there any relationship between the lengths of finite groups and In this paper we answer this question when at least one of the groups is a non-abelian 2-generated group. 1. Introduction G AAA be a AAAAA AOAAAAAIAAA AAOAA AhAAA A A 2 is an . A . -1 i i i 1 1 . . i 1 m 1 2 m 2. m 190 A 1 1 2 -y 0 1 2 . .-1 -2 .-3 . 0 1 . . -1 . -2 .G. . . . 2 . 2 . Proposition A. For every -generated non-abelian finite every cyclic order m . . 1 . Proposition B. For every -generated non-abelian finite . . V1 . 1 1 191 Proposition C. For a positive integer 2 let . 2 22 . .be non-abelian 2 -generated finite groups. For every 2 2 2 2 2 define A . 2 2 .12 2 .. as follows if. 2. if. . 12 if. . . Then ft ft ft ft ft ft ft ft ft ftftft ftft ft ft ft ftft ft ft 1 1 a 2 n . 1 . . A 1 . 2. 2. 1 2 1222 2n- 2 2-12 2 -1. Proposition D. . For every 2 2 and 2 . . 21 2 2. 2 . Forevery2 2 1 . 2. 2 2 .dụ. 1 1 -2 2. 2 .lị. and in general for an integer 2 and for every 2 . we define A . 2 2 12 2 2. 122 2 where 2 2 if m 2 22 2 . 2 22 if. 2 otherwise. Then . .22 2. 2 2 .2k. .4 . where . is the least positive integer such that for all values of 2. . . 22q. . . . Proposition E. For an integer 2 and for A . be as in the Proposition D. Then A 1 2n. 2. .