  
  [1X1 Introduction[0X
  
  
  [1X1.1 General aims[0X
  
  [5XLAGUNA[0m -- [12XL[0mie [12XA[0ml[12XG[0mebras and [12XUN[0mits of group [12XA[0mlgebras -- is the new name of the
  [5XGAP[0m4 package [5XLAG[0m. The [5XLAG[0m package arose as a byproduct of the third author's
  PhD  thesis  [Ros97].  Its  first version was ported to [5XGAP[0m4 and was brought
  into  the  standard  [5XGAP[0m4  package  format during his visit to St Andrews in
  September 1998.
  
  The  main objective of [5XLAG[0m is to deal with Lie algebras associated with some
  associative  algebras,  and,  in particular, Lie algebras of group algebras.
  Using  [5XLAG[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 [5XLAGUNA[0m the main part of the Lie algebra
  functionality is heavily built on the previous [5XLAG[0m releases.
  
  The  [5XGAP[0m4 package [5XLAGUNA[0m also extends the [5XGAP[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. [5XLAGUNA[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 [5XLAGUNA[0m provides a part of
  the  functionality  of  the  [5XSISYPHOS[0m program, which was developed by Martin
  Wursthorn to study the modular isomorphism problem; see [Wur93].
  
  The   corresponding  functions  of  [5XLAGUNA[0m  use  the  same  algorithmic  and
  theoretical  approach  as those in [5XSISYPHOS[0m. The reason why we reimplemented
  the  normalised unit group algorithms in the [5XLAGUNA[0m package is that [5XSISYPHOS[0m
  has  no  interface  to  [5XGAP[0m4, and, even in [5XGAP[0m3, it is cumbersome to use the
  [5XSISYPHOS[0m  output for further computation with the normalised unit group. For
  instance, using [5XSISYPHOS[0m with its [5XGAP[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 [5XLAGUNA[0m.
  
  
  [1X1.2 General computations in group rings[0X
  
  The  [5XLAGUNA[0m  package  provides  a  set  of functions to carry out some basic
  computations  with a group ring and its elements. Among other things, [5XLAGUNA[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  [5XLAGUNA[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 [14X4.1[0m--[14X4.3[0m for more details.
  
  
  [1X1.3 Computations in the normalized unit group[0X
  
  One of the aims of the [5XLAGUNA[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  [Sim94, Chapter 9]. For this computation we use an algorithm that
  was  also  used in the [5XSISYPHOS[0m package. For a brief description see Chapter
  [14X3[0m. The functions that compute the structure of the normalised unit group are
  described in Section [14X4.4[0m.
  
  
  [1X1.4 Computing Lie properties of the group algebra[0X
  
  The  functions  that  are  used to compute Lie properties of p-modular group
  algebras  were already included in the previous versions of [5XLAG[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 [Ros00] 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 [14X4.5[0m.
  
  
  [1X1.5 Installation and system requirements[0X
  
  [5XLAGUNA[0m  does  not  use  external  binaries  and,  therefore,  works  without
  restrictions  on the type of the operating system. It is designed for [5XGAP[0m4.4
  and no compatibility with previous releases of [5XGAP[0m4 is guaranteed.
  
  To  use  the  [5XLAGUNA[0m online help it is necessary to install the [5XGAP[0m4 package
  [5XGAPDoc[0m  by  Frank L\"ubeck and Max Neunh\"offer, which is available from the
  [5XGAP[0m site or from [7Xhttp://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/[0m.
  
  [5XLAGUNA[0m  is  distributed in standard formats ([11Xzoo[0m, [11Xtar.gz[0m, [11Xtar.bz2[0m, [11X-win.zip[0m)
  and  can  be obtained from [7Xhttp://www.cs.st-andrews.ac.uk/~alexk/laguna.htm[0m.
  To unpack the archive [11Xlaguna-X.X.X.zoo[0m you need the program [11Xunzoo[0m, which can
  be  obtained  from  the [5XGAP[0m homepage [7Xhttp://www.gap-system.org/[0m (see section
  `Distribution').   To  install  [5XLAGUNA[0m,  copy  this  archive  into  the  [11Xpkg[0m
  subdirectory  of  your  [5XGAP[0m4.4 installation. The subdirectory [11Xlaguna[0m will be
  created in the [11Xpkg[0m directory after the following command:
  
  [10Xunzoo -x laguna-X.X.X.zoo[0m
  
