Small derived quotients in finite p-groups

More than 70 years ago, P. Hall showed that if $G$ is a finite $p$-group such that a term \der G{d+1} of the derived series is non-trivial, then the order of the quotient \der Gd/\der G{d+1} is at least $p^{2^d+1}$. Recently Mann proved that, in a finite $p$-group, Hall's lower bound can be taken for at most two distinct $d$. We improve this result and show that if $p$ is odd, then it can only be taken for two distinct $d$ in a group with order $p^6$.Comment: Two related papers have been submitted. The material have been reorganised for Versions 2 and results migrated between paper

Computing Nilpotent Quotients in Finitely Presented Lie Rings

A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor of $L$. LieNQ's complexity is studied and it is shown that optimizing the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP 3.5 interface is available.Comment: DVI and Post-Script files onl

A computer-based approach to the classification of nilpotent Lie algebras

We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over \F_2 and those of dimension at most~7 over \F_3 and \F_5.Comment: submitte