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Michael Vaughan-Lee
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The restricted Burnside problem On counterexamples to the Hughes conjecture Graham Higman's PORC conjecture Lie relators in groups of exponent p
For many years most of my research was centred on the restricted Burnside problem: "Given positive integers m and n, is there a bound on the orders of finite m-generator groups of exponent n?" This question was answered in the affirmative by Efim Zel'manov in 1991. His result implies that there is a largest finite m-generator group of exponent n, R(m,n). But many questions remain unanswered for all but a few values of m and n. What is the order of R(m,n)? What is its structure? If we can't find the exact order, can we find a reasonable bound for the order? I am interested in applying both theoretical and computational techniques to answering these questions. Efim Zelmanov and I have found bounds for the order of R(m,n) for general m and n. (See #Upper bounds in the restricted Burnside problem, J. Algebra 162 (1993), 107-145, and #Upper bounds in the restricted Burnside problem II, International J. Algebra and Computation 6 (1996), 735-744.) And I have used computers to obtain information about groups of exponent 4, 5 and 7. See (for example) #The restricted Burnside problem, Oxford University Press, 1993, #Lie rings of groups of prime exponent, J. Austral. Math. Soc. 49 (1990), 386-398, #The nilpotency class of finite groups of exponent p, Trans. Amer. Math. Soc. 346 (1994), 617-640, #Engel-4 groups of exponent 5, Proc. London Math. Soc. 74 (1997), 306-334, (with M.F. Newman) #Engel-4 groups of exponent 5 II, Proc. London Math. Soc. (3) 79 (1999) 283-317, (with Eamonn O'Brien) The 2-generator restricted Burnside group of exponent 7, Internat J. of Algebra and Computation 12 (2002) 575-592.
The associated Lie rings of groups of exponent p are Engel-(p-1) Lie algebras over the field of p elements. By studying these Lie algebras we obtain information about the groups R(m,p). I have developed computer programs for computing Engel Lie algebras, and many of the group theoretic results proved in the publications listed above make use of computer calculations with Lie algebras. For example, Mike Newman and I have computed the free 2-generator Engel-6 Lie algebra over Z_7. It has class 29 and dimension 23789. We have also computed the free 2-generator Lie algebra in the variety of Lie algebras satisfying all the multilinear identities satisfied by the associated Lie rings of groups of exponent 7. It has class 29 and dimension 20418. (See http://www.ams.org/era/home-1998.html.) This implies that R(2,7) has class at most 29 and order at most 7^20418, though it turns out that R(2,7) has class 28 and order 7^20416. So the associated Lie ring of R(2,7) satisfies relations which are not consequences of multilinear Lie relators.
I have developed computer programs for studying varieties of superalgebras defined by multilinear identities, and have used these programs to obtain results in associative algebras, Lie algebras, and Jordan algebras. See for example #An algorithm for computing graded algebras, J. Symbolic Computation 16 (1993), 345-354 and #Superalgebras and dimensions of algebras, International J. Algebra and Computation 8 (1998), 97-125.
George Havas and I have proved that 4-Engel groups are locally nilpotent. Our proof uses the Knuth-Bendix string rewriting procedure to prove that a certain three generator 4-Engel group is locally nilpotent. Our paper has been accepted for publication in the International Journal of Algebra and Computation.
I recently got intrigued by the possibility of using the Lazard correspondence and the Baker-Campbell-Hausdorff formula to classify groups of order p^n by classifying nilpotent Lie rings of order p^n. Mike Newman, Eamonn O'Brien and I used this method to classify the groups of order p^6, and Eamonn and I used it to classify the groups of order p^7. We have constructed databases of the groups. For details see http://www.math.auckland.ac.nz/~obrien/research/p7.html.
George Havas and I have constructed power commutator presentations for various counterexamples to the Hughes conjecture. For a preprint and supporting MAGMA programs see our website.
I was reading Marcus du Sautoy's book "Finding Moonshine", and I came across a group presentation which Marcus said he wanted enscribed on his gravestone. I got interested in this group, which can be interpreted as a class two group of order p^9 and exponent p. Marcus and I managed to prove that the number of descendants of order p^10 of this group is not PORC. For a draft article on this, and for other articles and supporting material, see website.
I have been interested in Lie relators in varieties of groups for many years, especially in Burnside varieties and in Engel varieties. My book "The restricted Burnside problem" (OUP) gives a complete basis for the multilinear Lie relators which hold in the associated Lie rings of groups of prime power exponent. Tim Wall extended my methods in "Multilinear Lie relators for varieties of groups" (J. Algebra 157 (1993), 341--393), and showed how in any variety of groups you can obtain a basis for the multilinear Lie relators. His basis has been of immense benefit in the study of n-Engel groups. I recently revisited an old result of Sanov that the Lie relators of weight at most 2p-2 which hold in the associated Lie rings of groups of exponent p are all consequences of the identity px = 0 and the (p-1)-Engel identity. Sanov's proof of this result is written in Russian (of course!) and has not been translated into English. In addition, Sanov draws heavily on Hausdorff's 1906 paper on the Baker-Campbell-Hausdorff formula, and I was unable to obtain a copy of that paper. For those reasons, but mainly in an attempt to exorcise some demons that are rattling about in my skull, I have written up a proof of Sanov's theorem, together with a complete self contained proof of Hausdorff's expression for the BCH formula. If you are interested in Sanov's proof, or Hausdorff's formula, see my note. Perhaps the main reason I wrote up a proof of Sanov's Theorem is that it provides a very short, easy, computer free refutation of Seymour Bachmuth's absurd claim posted on the arXiv that the Burnside group B(2,q) if finite for all prime powers q. See note. |