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

Groups of prime-power order with a small second derived quotient

For odd primes we prove some structure theorems for finite $p$-groups $G$, such that $G''\neq 1$ and $|G'/G''|=p^3$. Building on results of Blackburn and Hall, it is shown that \lcs G3 is a maximal subgroup of $G'$, the group $G$ has a central decomposition into two simpler subgroups, and, moreover, $G'$ has one of two isomorphism types.Comment: 16 page

The isomorphism problem for universal enveloping algebras of nilpotent Lie algebras

In this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent Lie algebras. We prove that if the characteristic of the underlying field is not~2 or~3, then the isomorphism type of a nilpotent Lie algebra of dimension at most~6 is determined by the isomorphism type of its universal enveloping algebra. Examples show that the restriction on the characteristic is necessary