Michael Vaughan-Lee

The restricted Burnside problem

Preface Contents Notation 1. Basic concepts 2. The associated Lie ring of a group 3. Kostrikins Theorem 4. Razmyslovs theorem 5. Groups of exponent two, three, and six 6. Groups of exponent four 7. Groups of prime exponent 8. Groups of prime-power exponent 9. Zelmanovs Theorem Appendix A Appendix B Index

A nilpotent quotient algorithm for graded Lie rings

A nilpotent quotient algorithm for graded Lie rings of prime characteristic is described. Thealgorithm has been implemented and applications have been made to the investigation of the associated Lie rings of Burnside groups. New results about Lie rings and Burnside groups are presented. These include detailed information on groups of exponent 5 and 7 and their associated Lie rings.

Breadth and commutator subgroups of p-groups

This inequality has already been established in a number of special cases. It was shown to hold for groups of class 2 by I. M. Bride [I], and it was shown to hold for metabelian groups in [3]. It was confirmed for groups G satisfying b(G) 3 ~o!(G)~ 1 [d(G) being the number of elements in a minimal generating set for G] by James Wiegold [6], and it was confirmed for groups G satisfying b(G) < p in [4]. This paper grew out of work James Wiegold and I did on the generation of p-groups by elements of maximal breadth [4], and I am indebted to him for a very fruitful correspondence on this subject. It has also been conjectured that j G’ j = pc1/2tacG)cbcc)+1r can only occur when G has class 2, or when b(G) = 2 and G has class 3, and an examination of the proof of this theorem shows that this is indeed the case.

4-ENGEL GROUPS ARE LOCALLY NILPOTENT

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

An Algorithm for Computing Graded Algebras

In this article I describe an algorithm for computing finite dimensional graded algebras, and I describe an implementation of the algorithm. As an application of the algorithm, I investigate associative algebras satisfying the identity x4 = 0. I show that if A is an associative algebra over a field of characteristic zero, and if a4 = 0 for all a � A, then A10 = {0}.

Engel-4 Groups of Exponent 5

We show that if

Superalgebras and Dimensions of Algebras

G

THE 2-GENERATOR RESTRICTED BURNSIDE GROUP OF EXPONENT 7

is a group of exponent 5, and if

UPPER BOUNDS IN THE RESTRICTED BURNSIDE PROBLEM II

G

Bounds in the restricted Burnside problem

satisfies the Engel-4 identity