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 (International Journal of Algebra and Computation 15 (2005) 649-682) and for which a pdf file is available: locally nilpotent preprint. Details of input and output files for various machine computations in this paper together with a copy of the Knuth-Bendix program RKBP (kindly provided by its author, Charles Sims) are available. A preprint of a related paper Computing with 4-Engel groups (in Groups St Andrews 2005, London Mathematical Society Lecture Note Series 340, Cambridge University Press (2007), 457-474) is also available: computing preprint. This paper elaborates on explicit computer calculations which provided some of the motivation behind the proof of local nilpotence. In particular we give details on the hardest coset enumerations now required to show in a direct proof that 4-Engel p-groups are locally finite for 5 <= p <= 31. We provide a theoretical result which enables us to do requisite coset enumerations much better and we also give a new, tight bound on the class of 4-Engel 5-groups.

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.

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Last updated: 25 October 2009