  
  [1X4 [33X[0;0YAn example of [5XUnitLib[105X[101X[1X usage[133X[101X
  
  [33X[0;0YWe will finish with several examples of [5XUnitLib[105X usage to give an idea how to
  work with the package.[133X
  
  [33X[0;0YIn  the first example we retrieve from the library the normalized unit group
  of  the  group  algebra of the dihedral group of order 128 over the field of
  two  elements, compute its center and express one of its generators in terms
  of group algebra elements:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIdGroup(DihedralGroup(128));[127X[104X
    [4X[28X[ 128, 161 ][128X[104X
    [4X[25Xgap>[125X [27XV := PcNormalizedUnitGroupSmallGroup( 128, 161 );[127X[104X
    [4X[28X<pc group of size 170141183460469231731687303715884105728 with 127 generators>[128X[104X
    [4X[25Xgap>[125X [27XC := Center( V );           [127X[104X
    [4X[28X<pc group with 34 generators>  [128X[104X
    [4X[25Xgap>[125X [27Xgens := MinimalGeneratingSet( C );;[127X[104X
    [4X[25Xgap>[125X [27XKG := UnderlyingGroupRing( V );[127X[104X
    [4X[28X<algebra-with-one over GF(2), with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27Xf := NaturalBijectionToNormalizedUnitGroup( KG );;[127X[104X
    [4X[25Xgap>[125X [27Xgens[8]^f;[127X[104X
    [4X[28X(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f7+(Z(2)^0)*f3*f4+(Z(2)^0)*f3*f5+(Z(2)^[128X[104X
    [4X[28X0)*f3*f6+(Z(2)^0)*f3*f7+(Z(2)^0)*f4*f5+(Z(2)^0)*f4*f6+(Z(2)^0)*f4*f7+(Z(2)^[128X[104X
    [4X[28X0)*f3*f4*f5+(Z(2)^0)*f3*f4*f6+(Z(2)^0)*f3*f4*f7+(Z(2)^0)*f3*f5*f6+(Z(2)^[128X[104X
    [4X[28X0)*f3*f5*f7+(Z(2)^0)*f3*f6*f7+(Z(2)^0)*f4*f5*f6+(Z(2)^0)*f4*f5*f7+(Z(2)^[128X[104X
    [4X[28X0)*f4*f6*f7+(Z(2)^0)*f3*f4*f5*f6+(Z(2)^0)*f3*f4*f5*f7+(Z(2)^0)*f3*f4*f6*f7+([128X[104X
    [4X[28XZ(2)^0)*f3*f5*f6*f7+(Z(2)^0)*f4*f5*f6*f7+(Z(2)^0)*f3*f4*f5*f6*f7[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the second example we will check the conjecture about the coincidence of
  the lower and upper Lie nilpotency indices of the modular group algebras for
  all non-abelian groups of order 64.[133X
  
  [33X[0;0YIt is known that these indices coincide for [22Xp[122X-groups with [22Xp>3[122X [BP92], but in
  the general case the problem remains open.[133X
  
  [33X[0;0YThe  indices  [22Xt_L(G)[122X  and  [22Xt^L(G)[122X  can be computed using the [5XLAGUNA[105X package.
  While  the  upper Lie nilpotency index can be expressed only in terms of the
  underlying  group  [22XG[122X,  the  lower  Lie nilpotency index is determined by the
  formula  [22Xt_L(G)  =[122X  cl  [22XV(KG)  +  1[122X  [Du92], and can be computed immediately
  whenever [22XV(KG)[122X is known.[133X
  
  [33X[0;0YIn  the  program  below  we  enumerate  all  groups of size 64 and check the
  conjecture  (we do not exclude from consideration some particular cases when
  the  conjecture is known to be true for [22Xp=2[122X, because this is beyond the task
  of this manual).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xfor n in [ 1 .. NrSmallGroups( 64 ) ] do[127X[104X
    [4X[25X>[125X [27Xif not IsAbelian( SmallGroup( 64, n ) ) then[127X[104X
    [4X[25X>[125X [27X  V := PcNormalizedUnitGroupSmallGroup( 64, n );[127X[104X
    [4X[25X>[125X [27X  KG := UnderlyingGroupRing( V );[127X[104X
    [4X[25X>[125X [27X  if LieLowerNilpotencyIndex( KG ) <>[127X[104X
    [4X[25X>[125X [27X     LieUpperNilpotencyIndex( KG ) then[127X[104X
    [4X[25X>[125X [27X    Print( n," - counterexample !!! \n" );[127X[104X
    [4X[25X>[125X [27X    break;[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus, the test was finished without finding a counterexample.[133X
  
  [33X[0;0YIn  the  next  example we will answer the question about possible nilpotency
  classes  of  normalized  unit groups of modular group algebras of nonabelian
  groups of order 64:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xcl := [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor n in [ 1 .. NrSmallGroups( 64 ) ] do[127X[104X
    [4X[25X>[125X [27Xif not IsAbelian( SmallGroup( 64, n ) ) then[127X[104X
    [4X[25X>[125X [27X  V := PcNormalizedUnitGroupSmallGroup( 64, n );  [127X[104X
    [4X[25X>[125X [27X  AddSet( cl, NilpotencyClassOfGroup( V ) );[127X[104X
    [4X[25X>[125X [27Xfi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xcl;[127X[104X
    [4X[28X[ 2, 3, 4, 5, 6, 7, 8, 16 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWith  [5XUnitLib[105X  you  can  perform  the  computation  from the last example in
  several  hours on a modern computer. Without [5XUnitLib[105X you will spend the same
  time  to  compute  only  several  normalized unit groups [22XV(KG)[122X for groups of
  order 128 with the help of the [5XLAGUNA[105X package. Note that without [5XLAGUNA[105X such
  computation would not be feasible at all.[133X
  
