  
  [1X2 [33X[0;0YA sample calculation with [5XLAGUNA[105X[101X[1X[133X[101X
  
  [33X[0;0YBefore  explaining  the theory behind the [5XLAGUNA[105X package we present a sample
  calculation to show the reader what [5XLAGUNA[105X is able to compute. We will carry
  out some calculations in the group algebra of the dihedral group of order 16
  over the field of two elements. First we create this modular group algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XK := GF( 2 );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[25Xgap>[125X [27XG := DihedralGroup( 16 );[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XKG := GroupRing( K, G );[127X[104X
    [4X[28X<algebra-with-one over GF(2), with 4 generators>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group  algebra  [10XKG[110X  has  some properties and attributes that are direct
  consequences of its definition. These can be checked very quickly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIsGroupAlgebra( KG ); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPModularGroupAlgebra( KG );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsFModularGroupAlgebra( KG );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingGroup( KG );[127X[104X
    [4X[28X<pc group of size 16 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XLeftActingDomain( KG );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  [10XKG[110X  is  naturally  a  group  algebra,  the  information  provided  by
  [10XLeftActingDomain[110X can also be obtained using two other functions as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingRing( KG );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingField( KG );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  us  construct  a  certain element of the group algebra. For example, we
  take  a  minimal generating system of the group [10XG[110X and find the corresponding
  elements in [10XKG[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XMinimalGeneratingSet( G );[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27Xl := List( last, g -> g^Embedding( G, KG ) );[127X[104X
    [4X[28X[ (Z(2)^0)*f1, (Z(2)^0)*f2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we construct an element [10Xx[110X as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xa :=l[1]; b:=l[2]; # a and b are images of group generators in KG[127X[104X
    [4X[28X(Z(2)^0)*f1[128X[104X
    [4X[28X(Z(2)^0)*f2[128X[104X
    [4X[25Xgap>[125X [27Xe := One( KG );    # for convenience, we denote the identity by e[127X[104X
    [4X[28X(Z(2)^0)*<identity> of ...[128X[104X
    [4X[25Xgap>[125X [27Xx := ( e + a ) * ( e + b ); [127X[104X
    [4X[28X(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2  [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe may investigate some of the basic properties of our element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSupport( x );[127X[104X
    [4X[28X[ <identity> of ..., f1, f2, f1*f2 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficientsBySupport( x );[127X[104X
    [4X[28X[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ][128X[104X
    [4X[25Xgap>[125X [27XLength( x );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XTraceOfMagmaRingElement( x );[127X[104X
    [4X[28XZ(2)^0[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  can also calculate the augmentation of [10Xx[110X, which is defined as the sum of
  its coefficients.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XAugmentation( x );[127X[104X
    [4X[28X0*Z(2)[128X[104X
    [4X[25Xgap>[125X [27XIsUnit( KG, x );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  the  augmentation of [10Xx[110X is zero, [10Xx[110X is not invertible, but [10X1+x[110X is. This
  is again very easy to check.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xy := e + x;[127X[104X
    [4X[28X(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2[128X[104X
    [4X[25Xgap>[125X [27XIsUnit( KG, y );[127X[104X
    [4X[28Xtrue  [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XLAGUNA[105X can calculate the inverse of [10X1+x[110X very quickly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xy^-1;[127X[104X
    [4X[28X(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^[128X[104X
    [4X[28X0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f4+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f2*f3*f4+([128X[104X
    [4X[28XZ(2)^0)*f1*f2*f3*f4[128X[104X
    [4X[25Xgap>[125X [27Xy * y^-1;[127X[104X
    [4X[28X(Z(2)^0)*<identity> of ... [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  may  also  want  to check whether [10Xy[110X is symmetric, that is, whether it is
  invariant under the classical involution; or whether it is unitary, that is,
  whether the classical involution inverts [10Xy[110X. We find that [10Xy[110X is neither.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XInvolution( y );[127X[104X
    [4X[28X(Z(2)^0)*f1+(Z(2)^0)*f1*f2+(Z(2)^0)*f2*f3*f4[128X[104X
    [4X[25Xgap>[125X [27Xy = Involution( y );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetric( y );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xy * Involution( y );[127X[104X
    [4X[28X(Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f2*f3*f4  [128X[104X
    [4X[25Xgap>[125X [27XIsUnitary( y );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  calculate  some  important  ideals  of  [10XKG[110X.  First  we  obtain  the
  augmentation  ideal  which is the set of elements with augmentation zero. In
  our  case the augmentation ideal of [10XKG[110X coincides with the radical of [10XKG[110X, and
  this is taken into account in [5XLAGUNA[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XAugmentationIdeal( KG );[127X[104X
    [4X[28X<two-sided ideal in <algebra-with-one over GF(2), with 4 generators>,[128X[104X
    [4X[28X  (dimension 15)>[128X[104X
    [4X[25Xgap>[125X [27XRadicalOfAlgebra( KG ) = AugmentationIdeal( KG );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is  well-known  that  the augmentation ideal of [10XKG[110X is a nilpotent ideal.
  Using  Jennings' theory on dimension subgroups, we can obtain its nilpotency
  index  without  immediate  calculation of its powers. This is implemented in
  [5XLAGUNA[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XAugmentationIdealNilpotencyIndex( KG );[127X[104X
    [4X[28X9[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOn  the  other  hand,  we  can also calculate the powers of the augmentation
  ideal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xs := AugmentationIdealPowerSeries( KG );;[127X[104X
    [4X[25Xgap>[125X [27Xs[2];[127X[104X
    [4X[28X<algebra of dimension 13 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XList(s,Dimension);[127X[104X
    [4X[28X[ 15, 13, 11, 9, 7, 5, 3, 1, 0 ][128X[104X
    [4X[25Xgap>[125X [27XLength(s);[127X[104X
    [4X[28X9[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that the length of this list is exactly the nilpotency index of the
  augmentation ideal of [10XKG[110X.[133X
  
  [33X[0;0YNow  let's work with the unit group of [10XKG[110X. First we calculate the normalized
  unit  group,  which  is  the  set  of  elements  with  augmentation one. The
  generators  of  the  unit group are obtained as explained in Chapter [14X3[114X. This
  can  be  computed  very  quickly, but further computation with this group is
  very inefficient.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XV := NormalizedUnitGroup( KG );[127X[104X
    [4X[28X<group of size 32768 with 15 generators>   [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  order to make our computation in the normalised unit group efficient, we
  calculate a power-commutator presentation for this group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XW := PcNormalizedUnitGroup( KG );[127X[104X
    [4X[28X<pc group of size 32768 with 15 generators>[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XGAP[105X  has  many  efficient  and  practical  algorithms  for groups given by a
  power-commutator presentation. In order to use these algorithms to carry out
  computation  in  the  normalised  unit group, we need to set up isomorphisms
  between the outputs of [10XNormalizedUnitGroup[110X and [10XPcNormalizedUnitGroup[110X.[133X
  
  [33X[0;0YThe  first  isomorphism maps [10XNormalizedUnitGroup(KG)[110X onto the polycyclically
  presented  [10XPcNormalizedUnitGroup(PC)[110X.  Let's find the images of the elements
  of the group [10XG[110X in [10XW[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xt := NaturalBijectionToPcNormalizedUnitGroup( KG );[127X[104X
    [4X[28XMappingByFunction( <group of size 32768 with 15 generators>, <pc group of size\[128X[104X
    [4X[28X 32768 with 15 generators>, function( x ) ... end )[128X[104X
    [4X[25Xgap>[125X [27XImage(t) = W;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList( AsList( G ), x -> ( x^Embedding( G, KG ) )^t );[127X[104X
    [4X[28X[ <identity> of ..., f1, f2, f4, f8, f1*f2, f1*f4, f1*f8, f2*f4, f2*f8, [128X[104X
    [4X[28X  f4*f8, f1*f2*f4, f1*f2*f8, f1*f4*f8, f2*f4*f8, f1*f2*f4*f8 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe second isomorphism is the inverse of the first.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xf := NaturalBijectionToNormalizedUnitGroup( KG );;[127X[104X
    [4X[25Xgap>[125X [27XImage(f) = V;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor example, we may calculate the conjugacy classes of the group [10XW[110X, and then
  map their representatives back into the group algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xcc := ConjugacyClasses( W );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cc );[127X[104X
    [4X[28X848[128X[104X
    [4X[25Xgap>[125X [27XRepresentative( cc[ Length( cc ) ] );[127X[104X
    [4X[28Xf1*f2*f3*f6*f10*f13[128X[104X
    [4X[25Xgap>[125X [27Xlast^f;[127X[104X
    [4X[28X(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^[128X[104X
    [4X[28X0)*f2*f3+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f3*f4[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YHaving  a power-commutator presentation of the normalised unit group, we may
  use  the  full  power of the [5XGAP[105X functionality for such groups. For example,
  the lower central series can be calculated very quickly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XLowerCentralSeries( W );[127X[104X
    [4X[28X[ <pc group of size 32768 with 15 generators>, [128X[104X
    [4X[28X  Group([ f4*f8, f5*f7*f11*f13*f15, f6*f7*f9*f11*f13*f14*f15, f8, f9*f13, [128X[104X
    [4X[28X      f10*f11, f12*f13, f13*f15, f14*f15 ]), [128X[104X
    [4X[28X  Group([ f8, f9*f15, f10*f11, f12*f15, f13*f15, f14*f15 ]), [128X[104X
    [4X[28X  Group([ f12*f15, f13*f15, f14*f15 ]), Group([ <identity> of ... ]) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet's  now  compute,  for  instance,  a  minimal system of generators of the
  centre  of  the normalised unit group. First we carry out the computation in
  the  group which is determined by the power-commutator presentation, then we
  map the result into our group algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XC := Centre( W );;[127X[104X
    [4X[25Xgap>[125X [27Xm := MinimalGeneratingSet( C );[127X[104X
    [4X[28X[ f8*f13*f14*f15, f13*f14*f15, f8*f12*f14*f15, f15, f4*f6*f8*f13 ][128X[104X
    [4X[25Xgap>[125X [27XList( m, g -> g^f );[127X[104X
    [4X[28X[ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^0)*f3*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f2*f3*f4, [128X[104X
    [4X[28X  (Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^0)*f3*f4+(Z(2)^0)*f1*f2*f3+([128X[104X
    [4X[28X    Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*<identity> of ...+(Z(2)^[128X[104X
    [4X[28X    0)*f1+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f3*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f3*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+([128X[104X
    [4X[28X    Z(2)^0)*f3*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f3*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f2*f3*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+([128X[104X
    [4X[28X    Z(2)^0)*f4+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f3*f4+(Z(2)^[128X[104X
    [4X[28X    0)*f1*f3*f4+(Z(2)^0)*f2*f3*f4 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  finish  our  example  by  calculating some properties of the Lie algebra
  associated with [10XKG[110X. This example needs no further explanation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XL := LieAlgebra( KG );[127X[104X
    [4X[28X#I  LAGUNA package: Constructing Lie algebra ...[128X[104X
    [4X[28X<Lie algebra of dimension 16 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XD := LieDerivedSubalgebra( L );[127X[104X
    [4X[28X#I  LAGUNA package: Computing the Lie derived subalgebra ...[128X[104X
    [4X[28X<Lie algebra of dimension 9 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XLC := LieCentre( L );[127X[104X
    [4X[28X<Lie algebra of dimension 7 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XLieLowerNilpotencyIndex( KG );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XLieUpperNilpotencyIndex( KG );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XIsLieAbelian( L );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsLieSolvable( L );[127X[104X
    [4X[28X#I  LAGUNA package: Checking Lie solvability ...[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLieMetabelian( L );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsLieCentreByMetabelian( L );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
