4-Engel groups; supplementary materials
Via this web page you have access to supplementary materials for the paper 4-Engel groups are locally nilpotent written by George Havas and M.R. Vaughan-Lee (for which a pdf preprint is available). Details of input and output files for various machine computations together with a copy of the Knuth-Bendix program RKBP (kindly provided by its author, Charles Sims) are available.

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]
@
quit
```
The 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
summary
```
This shows that T has class 4.

Last updated: 13 July 2004