1. Introduction
  
  
  1.1 General aims
  
  LAGUNA -- Lie AlGebras and UNits of group Algebras -- is the new name of the
  GAP4 package LAG. The LAG package arose as a byproduct of the third author's
  PhD  thesis [R97]. Its first version was ported to GAP4 and was brought into
  the standard GAP4 package format during his visit to St Andrews in September
  1998.
  
  The  main objective of LAG is to deal with Lie algebras associated with some
  associative  algebras,  and,  in particular, Lie algebras of group algebras.
  Using  LAG it is possible to verify some properties or calculate certain Lie
  ideals  of  such  Lie  algebras  very  efficiently,  due  to  their  special
  structure. In the current version of LAGUNA the main part of the Lie algebra
  functionality is heavily built on the previous LAG releases.
  
  The  GAP4 package LAGUNA also extends the GAP functionality for calculations
  with units of modular group algebras. In particular, using this package, one
  can  check  whether an element of such a group algebra is invertible. LAGUNA
  also  contains  an implementation of an efficient algorithm to calculate the
  (normalized)  unit  group  of the group algebra of a finite p-group over the
  field  of p elements. Thus, the present version of LAGUNA provides a part of
  the  functionality  of  the  SISYPHOS program, which was developed by Martin
  Wursthorn to study the modular isomorphism problem; see [W93].
  
  The   corresponding  functions  of  LAGUNA  use  the  same  algorithmic  and
  theoretical  approach  as those in SISYPHOS. The reason why we reimplemented
  the  normalised unit group algorithms in the LAGUNA package is that SISYPHOS
  has  no  interface  to  GAP4, and, even in GAP3, it is cumbersome to use the
  SISYPHOS  output for further computation with the normalised unit group. For
  instance, using SISYPHOS with its GAP3 interface, it is difficult to embed a
  finite  p-group into the normalized unit group of its group algebra over the
  field of p elements, but this can easily be done with LAGUNA.
  
  
  1.2 General computations in group rings
  
  The  LAGUNA  package  provides  a  set  of functions to carry out some basic
  computations  with a group ring and its elements. Among other things, LAGUNA
  provides  elementary  functions  to  compute  such basic notions as support,
  length,  trace and augmentation of an element. For modular group algebras of
  finite  p-groups  LAGUNA  is  able  to  calculate the power-structure of the
  augmentation  ideal,  which is useful for the construction of the normalised
  unit group; see Sections 4.1--4.3 for more details.
  
  
  1.3 Computations in the normalized unit group
  
  One of the aims of the LAGUNA package is to carry out efficient computations
  in  the  normalised unit group of the group algebra FG of a finite p-group G
  over the field F of p elements. If U is the unit group of FG then it is easy
  to  see  that  U  is  the  direct product of F^* and V(FG), where F^* is the
  multiplicative  group  of  F,  and V(FG) is the group of normalised units. A
  unit of FG of the form alpha_1 * g_1 + alpha_2 * g_2 + cdots + alpha_k * g_k
  with alpha_i in F and g_i in G is said to be normalised if the sum alpha_1 +
  alpha_2 + cdots + alpha_k is equal to 1.
  
  It  is  well-known that the normalised unit group V has order |F|^|G|-1, and
  so  V  is  a finite p-group. Thus computing V efficiently means to compute a
  polycyclic  presentation  for  V. For the theory of polycyclic presentations
  refer to [S94, Chapter 9]. For this computation we use an algorithm that was
  also  used  in the SISYPHOS package. For a brief description see Chapter 3..
  The  functions  that  compute the structure of the normalised unit group are
  described in Section 4.4.
  
  
  1.4 Computing Lie properties of the group algebra
  
  The  functions  that  are  used to compute Lie properties of p-modular group
  algebras  were already included in the previous versions of LAG. The bracket
  operation  [*,*]  on a p-modular group algebra FG is defined by [a,b]=ab-ba.
  It  is  well-known  and  very  easy  to  check  that (FG, +, [*,*]) is a Lie
  algebra.  Then  we may ask what kind of Lie algebra properties are satisfied
  by  FG.  The  results  in  [LR86],  [PPS73],  and [R00] give fast, practical
  algorithms  to  check  whether  the  Lie  algebra  FG is abelian, nilpotent,
  soluble,  centre-by-metabelian,  etc.  The  functions  that  implement these
  algorithms are described in Section 4.5.
  
  
  1.5 Installation and system requirements
  
  LAGUNA  does  not  use  external  binaries  and,  therefore,  works  without
  restrictions  on the type of the operating system. It is designed for GAP4.4
  and no compatibility with previous releases of GAP4 is guaranteed.
  
  To  use  the  LAGUNA online help it is necessary to install the GAP4 package
  GAPDoc  by  Frank L\"ubeck and Max Neunh\"offer, which is available from the
  GAP site or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.
  
  LAGUNA  is  distributed in standard formats (zoo, tar.gz, tar.bz2, -win.zip)
  and  can  be obtained from http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm.
  To  unpack  the archive laguna-3.4.zoo you need the program unzoo, which can
  be  obtained  from  the GAP homepage http://www.gap-system.org/ (see section
  `Distribution').   To  install  LAGUNA,  copy  this  archive  into  the  pkg
  subdirectory  of  your  GAP4.4 installation. The subdirectory laguna will be
  created in the pkg directory after the following command:
  
  unzoo -x laguna-3.4.zoo