  
  [1m[4m[31m1. Introduction[0m
  
  
  [1m[4m[31m1.1 General aims[0m
  
  [1mLAGUNA[0m -- [1m[46mL[0mie [1m[46mA[0ml[1m[46mG[0mebras and [1m[46mUN[0mits of group [1m[46mA[0mlgebras -- is the new name of the
  [1mGAP[0m4 package [1mLAG[0m. The [1mLAG[0m package arose as a byproduct of the third author's
  PhD  thesis [R97]. Its first version was ported to [1mGAP[0m4 and was brought into
  the standard [1mGAP[0m4 package format during his visit to St Andrews in September
  1998.
  
  The  main objective of [1mLAG[0m is to deal with Lie algebras associated with some
  associative  algebras,  and,  in particular, Lie algebras of group algebras.
  Using  [1mLAG[0m 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 [1mLAGUNA[0m the main part of the Lie algebra
  functionality is heavily built on the previous [1mLAG[0m releases.
  
  The  [1mGAP[0m4 package [1mLAGUNA[0m also extends the [1mGAP[0m 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. [1mLAGUNA[0m
  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 [1mLAGUNA[0m provides a part of
  the  functionality  of  the  [1mSISYPHOS[0m program, which was developed by Martin
  Wursthorn to study the modular isomorphism problem; see [W93].
  
  The   corresponding  functions  of  [1mLAGUNA[0m  use  the  same  algorithmic  and
  theoretical  approach  as those in [1mSISYPHOS[0m. The reason why we reimplemented
  the  normalised unit group algorithms in the [1mLAGUNA[0m package is that [1mSISYPHOS[0m
  has  no  interface  to  [1mGAP[0m4, and, even in [1mGAP[0m3, it is cumbersome to use the
  [1mSISYPHOS[0m  output for further computation with the normalised unit group. For
  instance, using [1mSISYPHOS[0m with its [1mGAP[0m3 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 [1mLAGUNA[0m.
  
  
  [1m[4m[31m1.2 General computations in group rings[0m
  
  The  [1mLAGUNA[0m  package  provides  a  set  of functions to carry out some basic
  computations  with a group ring and its elements. Among other things, [1mLAGUNA[0m
  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  [1mLAGUNA[0m  is  able  to  calculate the power-structure of the
  augmentation  ideal,  which is useful for the construction of the normalised
  unit group; see Sections [1m4.1[0m--[1m4.3[0m for more details.
  
  
  [1m[4m[31m1.3 Computations in the normalized unit group[0m
  
  One of the aims of the [1mLAGUNA[0m 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 [1mSISYPHOS[0m package. For a brief description see Chapter [1m3.[0m.
  The  functions  that  compute the structure of the normalised unit group are
  described in Section [1m4.4[0m.
  
  
  [1m[4m[31m1.4 Computing Lie properties of the group algebra[0m
  
  The  functions  that  are  used to compute Lie properties of p-modular group
  algebras  were already included in the previous versions of [1mLAG[0m. 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 [1m4.5[0m.
  
  
  [1m[4m[31m1.5 Installation and system requirements[0m
  
  [1mLAGUNA[0m  does  not  use  external  binaries  and,  therefore,  works  without
  restrictions  on the type of the operation system. It is designed for [1mGAP[0m4.4
  and no compatibility with previous releases of [1mGAP[0m4 is guaranteed.
  
  To  use  the  [1mLAGUNA[0m online help it is necessary to install the [1mGAP[0m4 package
  [1mGAPDoc[0m  by  Frank L\"ubeck and Max Neunh\"offer, which is available from the
  [1mGAP[0m site or from [34mhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[0m.
  
  [1mLAGUNA[0m  is  distributed in standard formats ([1mzoo[0m, [1mtar.gz[0m, [1mtar.bz2[0m, [1m-win.zip[0m)
  and  can  be  obtained from [34mhttp://ukrgap.exponenta.ru/laguna.htm[0m. To unpack
  the  archive  [1mlaguna-3.3.3.zoo[0m  you  need  the  program  [1munzoo[0m, which can be
  obtained  from  the  [1mGAP[0m  homepage  [34mhttp://www.gap-system.org/[0m  (see section
  `Distribution').   To  install  [1mLAGUNA[0m,  copy  this  archive  into  the  [1mpkg[0m
  subdirectory  of  your  [1mGAP[0m4.4 installation. The subdirectory [1mlaguna[0m will be
  created in the [1mpkg[0m directory after the following command:
  
  [22m[32munzoo -x laguna-3.3.3.zoo[0m
  
