The crucial Knuth-Bendix computation: proving T
is nilpotent.
In a directory which includes the three files
final.rl1,
final.sys and
final.sb1,
run rkbp using lenlex ordering with the following commands:
input final summary kb 10 2 10 10 summary add_engel 4 2 0 5 10 10 summary kb 10 1 10 10 summary rewrite_words none names subword x11x11 [x10,x1,x1] [x10,x1,x2] [x10,x1,x3] [x10,x3] [x9,x3] [x7,x2] [x6,x2] [x4,x2] @ quitThe output of this run gives us the following relations: x11x11 = [x10,x1,x2] = [x10,x1,x3] = [x10,x3] = 1; [x10,x1,x1] = x11; [x9,x3] = x1x10X1X10; [x7,x2] = x10x10; [x6,x2] = x3x9X3; and [x4,x2] = x8x10. From these we can deduce that T is nilpotent.
Alternatively, we can observe the following relations: [x11,x1] = [x11,x2] = [x11,x3] = [x10,x1,x2] = [x10,x1,x3] = [x10,x3] = 1; [x10,x1,x1] = x11; [x9,x3] = x1x10X1X10; [x7,x2] = x10x10; [x6,x2] = x3x9X3; and [x4,x2] = x8x10. Adding these relations to T we can obtain a confluent rewriting system by now doing a Knuth-Bendix computation with a wreath product ordering. In a directory which includes the three files check.rl1, check.sys and check.sb1, run rkbp with the following input commands:
input check summary kb 10 -1 4 50 summary add_engel 4 3 0 5 4 50 summaryThis shows that T has class 4. Last updated: 13 July 2004