M. F. Newman

GRADED LIE ALGEBRAS OF MAXIMAL CLASS

We study graded Lie algebras of maximal class over a field

A nilpotent quotient algorithm for graded Lie rings

\mathbf {F}

The groups of order 128

of positive characteristic

Classifying 2-groups by coclass

p

On critical groups

. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct

Thin groups of prime-power order and thin line algebras

| \mathbf {F}|^{\aleph _{0}}

Cross varieties of groups

pairwise non-isomorphic such algebras, and

Calculating presentations for certain kinds of quotient groups

\max \{| \mathbf {F}|, \aleph _{0} \}

On non-cross varieties of groups

soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role

Just non-finitely-based varieties of groups

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.