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Michael Vaughan-Lee
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Lie rings of groups of prime exponent The restricted Burnside problem Upper bounds in the restricted Burnside problem An algorithm for computing graded algebras The nilpotency class of finite groups of exponent p Upper bounds in the restricted Burnside problem II Superalgebras and dimensions of algebras Engel-4 groups of exponent 5 II Generating matrices for R(2,5) 4-Engel groups are locally nilpotent
I show that the associated Lie ring of the largest finite 3-generator group of exponent 5 has dimension 2282, and that it is a free Lie algebra in the variety of Lie algebras determined by the multilinear identities satisfied by the associated Lie rings of groups of exponent 5. This implies that the largest finite 3-generator group of exponent 5 has order 5^2282, and nilpotency class 17.
This book aims to give a unified treatment of the restricted Burnside problem using the theory of the multilinear identities which hold in the associated Lie rings of groups of prime-power exponent. It contains versions of Kostrikin's and Zel'manov's solutions of the restricted Burnside problem for prime and prime-power exponents, as well as a version of Razmyslov's proof of the non-solvability of locally finite groups of exponent p^k for p^k > 3. The book also contains detailed information about groups of exponent 4,5,6,7,8, and 9.
We show that a finite m-generator group (m > 1) of prime-power exponent q = p^k has order at most m^m^...^m where the height of this tower is q^q^q. Although this bound was intended as a bit of a joke, M.F. Newman has pointed out that bounds of this sort are not unrealistic. Our theorem could be stated as: a finite m-generator group of prime power exponent q has order at most 2^2^...^2^m, where the height of the tower is q^q^q. If we let F be the free group of rank m, and define normal subgroups F = N_0 > N_1 > N_2 > ... by setting N_{i+1} = (N_i)^2, then F/N_k is an m-generator group of exponent 2^k. A simple induction using Schreier's formula for the rank of a subgroup of F shows that F/N_k has order greater than 2^2^...^2^m, where there are k twos in the tower.
I describe an algorithm for computing finite dimensional graded superalgebras. As an application of the algorithm I show that if A is an associative algebra over a field of characteristic zero, and if A satisfies the identity x^4 = 0 for all x in A, then A^10 = 0.
I investigate the properties of Lie algebras of characteristic p which satisfy the Engel-n identity for some n < p. I establish a criterion which (when satisfied) implies that if a and b are elements of an Engel-n Lie algebra L then ab^(n-2) generates a nilpotent ideal of L. I show that this criterion is satisfied for n = 6 and p = 7, and deduce that if G is a finite m-generator group of exponent 7, then G is nilpotent of class at most 51m^8. It is an open question whether there is a bound for the class of G which is linear in m. A bound of 15m seems possible. In the case of finite m-generator groups of exponent 5 the class is bounded by 6m. On the other hand, the argument given above implies that the class of an m-generator group of exponent 8 cannot be bounded by a polynomial in m.
We show that a finite m-generator group (m > 1) of exponent n has order at most m^m^...^m where the height of this tower is n^n^n. The proof uses the classification of finite simple groups, together with the Hall-Higman reduction of the restricted Burnside problem to the case of prime-power exponent.
I prove that Engel-4 groups of exponent 5 are locally finite, and that a group of exponent 5 is locally finite if and only if its 3-generator subgroups are finite. The article combines traditional proof with numerous computer calculations using coset enumeration and the p-quotient algorithm.
Several authors have used the representation theory of symmetric groups and superalgebras to prove that certain classes of algebras are nilpotent. We show how to extend these techniques to facilitate the computation of the dimensions of relatively free algebras, and we prove two general theorems which formalize these techniques. The ideas described here have been used in the computation of the dimensions of the associated Lie rings of free Engel-4 groups of exponent 5.
We find the exact order and class of the free Engel-4 groups of exponent 5. We show that their associated Lie rings are relatively free. I proved in an earlier paper that Engel-4 groups of exponent 5 are locally finite. So if we let G_m be the free m-generator Engel-4 group of exponent 5, then G_m is finite. We show that G_m has order 5^{h_m} where h_m = m + Sum_{k in [2..m]} m-choose-k (g_k + c_k) where g_k = (k-1)f_2k + (k+1)f_{2k-1}, with f_r the r-th Fibonacci number, and where c_k = 0 for k > 10 and c_2=3, c_3=87, c_4 = 595, c_5 = 1851, c_6 = 2996, c_7 = 2562, c_8 = 1094, c_9 = 224, c_10 = 35. We also show that G_2 has class 6, G_3 has class 8, and that G_m has class 2m for m >3.
The largest finite 2-generator group of exponent 5, R(2,5), has order 5^34 and nilpotency class 12. It can be embedded in the group of 66x66 upper unitriangular matrices over GF(5). The Magma program r25mats.txt defines two 66x66 matrices a and b which generate R(2,5). If you run the algebraic programming language Magma, and load in this file then you can inspect these matrices, multiply them, and so on, with Magma. The program defines a Magma function c(x,y) which computes the commutator [x,y] for any two (non-singular) 66x66 matrices over GF(5), and also defines a logical function isr25(x,y) which returns the value true if x and y generate a copy of R(2,5), and returns the value false otherwise. |