805 research outputs found
Small derived quotients in finite p-groups
More than 70 years ago, P. Hall showed that if is a finite -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 . Recently Mann
proved that, in a finite -group, Hall's lower bound can be taken for at most
two distinct . We improve this result and show that if is odd, then it
can only be taken for two distinct in a group with order .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 . Provided that the Lie ring is graded, it is
possible to determine the canonical presentation for a lower central factor of
. 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 -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 -groups ,
such that and . Building on results of Blackburn and
Hall, it is shown that \lcs G3 is a maximal subgroup of , the group
has a central decomposition into two simpler subgroups, and, moreover, 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
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