<!-- $Id: example.xml,v 1.13 2003/06/22 17:11:17 alexk Exp $ -->
<Chapter Label="Example">
<Heading>A sample calculation with &LAGUNA;</Heading>

Before explaining the theory behind the &LAGUNA; package we present 
a sample calculation to show the reader what &LAGUNA; 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.

<Example>
<![CDATA[
gap> K := GF( 2 );
GF(2)
gap> G := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>
gap> KG := GroupRing( K, G );
<algebra-with-one over GF(2), with 4 generators>
]]>
</Example>

The group algebra <Code>KG</Code> has some properties and attributes 
that are direct consequences of its definition. These can be checked 
very quickly.

<Example>
<![CDATA[
gap> IsGroupAlgebra( KG ); 
true
gap> IsPModularGroupAlgebra( KG );
true
gap> IsFModularGroupAlgebra( KG );
true
gap> UnderlyingGroup( KG );
<pc group of size 16 with 4 generators>
gap> LeftActingDomain( KG );
GF(2)
]]>
</Example>

Since <Code>KG</Code> is naturally a group algebra, the information provided by
<Code>LeftActingDomain</Code> can also be obtained using two other functions 
as follows.

<Example>
gap> UnderlyingRing( KG );
GF(2)
gap> UnderlyingField( KG );
GF(2)
</Example>

Let us construct a certain element of the group algebra.
For example, we take a minimal generating system of
the group <Code>G</Code> and find the corresponding elements in
<Code>KG</Code>.

<Example>
<![CDATA[
gap> MinimalGeneratingSet( G );
[ f1, f2 ]
gap> l := List( last, g -> g^Embedding( G, KG ) );
[ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
]]>
</Example>

Now we construct an element <Code>x</Code> as follows.

<Example>
<![CDATA[
gap> a :=l[1]; b:=l[2]; # a and b are images of group generators in KG
(Z(2)^0)*f1
(Z(2)^0)*f2
gap> e := One( KG );    # for convenience, we denote the identity by e
(Z(2)^0)*<identity> of ...
gap> x := ( e + a ) * ( e + b ); 
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2  
]]>
</Example>

We may investigate some of the basic properties of our element.

<Example>
<![CDATA[
gap> Support( x );
[ <identity> of ..., f1, f2, f1*f2 ]
gap> CoefficientsBySupport( x );
[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ]
gap> Length( x );
4
gap> TraceOfMagmaRingElement( x );
Z(2)^0
]]>
</Example>

We can also calculate the augmentation of <Code>x</Code>, which is defined as
the sum of its coefficients.

<Example>
gap> Augmentation( x );
0*Z(2)
gap> IsUnit( KG, x );
false
</Example>

Since the  augmentation of <Code>x</Code> 
is zero, <Code>x</Code> is not invertible, but
<Code>1+x</Code> is. This is again very easy to check.

<Example>
<![CDATA[
gap> y := e + x;
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> IsUnit( KG, y );
true  
]]>
</Example>

&LAGUNA; can calculate the inverse of <Code>1+x</Code> very quickly.

<Example>
<![CDATA[
gap> y^-1;
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(
Z(2)^0)*f1*f4+(Z(2)^0)*f2*f4+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f2*f3*f4+(Z(2)^
0)*f1*f2*f3*f4
gap> y * y^-1;
(Z(2)^0)*<identity> of ... 
]]>
</Example>

We may also want to  check whether <Code>y</Code> is symmetric, that is, 
whether it is invariant under the classical involution; or whether it is
unitary, that is, whether the classical involution inverts <Code>y</Code>.
We find that <Code>y</Code> is neither.

<Example>
<![CDATA[
gap> Involution( y );
(Z(2)^0)*f1+(Z(2)^0)*f1*f2+(Z(2)^0)*f2*f3*f4
gap> y = Involution( y );
false
gap> IsSymmetric( y );
false
gap> y * Involution( y );
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f2*f3*f4  
gap> IsUnitary( y );
false
]]>
</Example>

Now we calculate some 
important ideals of <Code>KG</Code>. First we obtain the augmentation
ideal which is the set of elements with augmentation zero. In our case the 
augmentation ideal of <Code>KG</Code> coincides with the radical of 
<Code>KG</Code>, and this is taken into account in &LAGUNA;.

<Example>
<![CDATA[
gap> AugmentationIdeal( KG );
<two-sided ideal in <algebra-with-one of dimension 16 over GF(2)>, 
  (dimension 15)>
gap> RadicalOfAlgebra( KG ) = AugmentationIdeal( KG );
true
]]>
</Example>

It is well-known that the augmentation ideal of <Code>KG</Code> 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 &LAGUNA;.

<Example>
gap> AugmentationIdealNilpotencyIndex( KG );
9
</Example>

<Alt Only="LaTeX">\newpage</Alt>  

On the other hand, we can also calculate the powers of the augmentation ideal.

<Example>
<![CDATA[
gap> AugmentationIdealPowerSeries( KG );
[ <algebra of dimension 15 over GF(2)>, <algebra of dimension 13 over GF(2)>, 
  <algebra of dimension 11 over GF(2)>, <algebra of dimension 9 over GF(2)>, 
  <algebra of dimension 7 over GF(2)>, <algebra of dimension 5 over GF(2)>, 
  <algebra of dimension 3 over GF(2)>, <algebra of dimension 1 over GF(2)>, 
  <algebra over GF(2)> ]
]]>
</Example>

We see that the length of this list is exactly the nilpotency index
of the augmentation ideal of <Code>KG</Code>.
<P/>

Now let's work with the unit group of <Code>KG</Code>.
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 <Ref Chap="Theory"/>. 
This can be computed very quickly, but further computation with this group 
is very inefficient.

<Example>
<![CDATA[
gap> V := NormalizedUnitGroup( KG );
<group of size 32768 with 15 generators>   
]]>
</Example>

In order to make our computation in the normalised unit group efficient,  
we calculate a power-commutator presentation for this group. 

<Example>
<![CDATA[
gap> W := PcNormalizedUnitGroup( KG );
<pc group of size 32768 with 15 generators>;
]]>
</Example>

&GAP; 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 <Code>NormalizedUnitGroup</Code>
and <Code>PcNormalizedUnitGroup</Code>.
<P/>

The first isomorphism maps <Code>NormalizedUnitGroup(KG)</Code>
onto the polycyclically presented <Code>PcNormalizedUnitGroup(PC)</Code>.
Let's find the images of the elements of the group <Code>G</Code> 
in <Code>W</Code>. 

<Example>
<![CDATA[
gap> t := NaturalBijectionToPcNormalizedUnitGroup( KG );
MappingByFunction( <group of size 32768 with 15 generators>, <pc group of size
32768 with 15 generators>, function( x ) ... end )
gap> List( AsList( G ), x -> ( x^Embedding( G, KG ) )^t );
[ <identity> of ..., f2, f1, f3, f7, f1*f2*f3, f2*f3, f2*f7, f1*f3, f1*f7, f3*f7,
  f1*f2*f7, f1*f2*f3*f7, f2*f3*f7, f1*f3*f7, f1*f2 ]   
]]>
</Example>

<Alt Only="LaTeX">\newpage</Alt>

The second isomorphism is the inverse of the first.

<Example>
gap> f := NaturalBijectionToNormalizedUnitGroup( KG );
[ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13, f14, f15 ] ->
[ (Z(2)^0)*f2, (Z(2)^0)*f1, (Z(2)^0)*f3, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2,
  (Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f2*f3, (Z(2)^0)*f1+(Z(2)^0)*f3+(Z(2)^0)*f1*f3,
  (Z(2)^0)*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(
    Z(2)^0)*f2*f3+(Z(2)^0)*f1*f2*f3, (Z(2)^0)*f2+(Z(2)^0)*f4+(Z(2)^0)*f2*f4,
  (Z(2)^0)*f1+(Z(2)^0)*f4+(Z(2)^0)*f1*f4, (Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f3*f4,
  (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f4+(Z(2)^
    0)*f2*f4+(Z(2)^0)*f1*f2*f4, (Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^
    0)*f2*f3+(Z(2)^0)*f2*f4+(Z(2)^0)*f3*f4+(Z(2)^0)*f2*f3*f4,
  (Z(2)^0)*f1+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^
    0)*f3*f4+(Z(2)^0)*f1*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(
    Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+(
    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)^
    0)*f2*f3*f4+(Z(2)^0)*f1*f2*f3*f4 ]  
</Example>

For example, we may calculate the conjugacy classes of the group <Code>W</Code>, 
and then map their representatives back into the group algebra.

<Example>
<![CDATA[
gap> cc := ConjugacyClasses( W );;
gap> Length( cc );
848
gap> Representative( cc[ Length( cc ) ] );
f1*f2*f4*f6*f12*f15
gap> last^f;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(
Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+(Z(2)^0)*f3*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^
0)*f1*f3*f4   
]]>
</Example>

Having a power-commutator presentation of the normalised unit group, we may 
use the full power of the &GAP; functionality for such groups. For example, the
lower central series can be calculated very quickly.

<Example>
<![CDATA[
gap> LowerCentralSeries( W );
[ <pc group of size 32768 with 15 generators>,
  Group([ f3, f5*f8*f10*f12*f13*f14*f15, f6*f8*f12*f14*f15, f7, f9*f12, f10*f14,
      f11*f13, f13*f14, f14*f15 ]),
  Group([ f7, f9*f12, f10*f15, f11*f15, f13*f15, f14*f15 ]),
  Group([ f11*f15, f13*f15, f14*f15 ]), Group([ <identity> of ... ]) ]  
]]>
</Example>

<Alt Only="LaTeX">\newpage</Alt>

Let'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.

<Example>
<![CDATA[
gap> C := Centre( W );
Group([ f3*f5*f13*f15, f7, f15, f13*f15, f14*f15, f11*f13*f14*f15 ])
gap> m := MinimalGeneratingSet( C );
[ f11, f13, f14, f15, f3*f5*f13*f15 ]
gap> List( m, g -> g^f );
[ (Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f3*f4, (Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(
    Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+(Z(2)^0)*f3*f4+(Z(2)^0)*f2*f3*f4,
  (Z(2)^0)*f1+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^
    0)*f3*f4+(Z(2)^0)*f1*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(
    Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+(
    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)^
    0)*f2*f3*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^
    0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^
    0)*f1*f2*f4+(Z(2)^0)*f1*f3*f4+(Z(2)^0)*f1*f2*f3*f4 ]    
]]>
</Example>

We  finish our example by calculating some properties of the  
Lie algebra associated with <Code>KG</Code>. 
This example needs no further explanation.

<Example>
<![CDATA[
gap> L := LieAlgebra( KG );
<Lie algebra of dimension 16 over GF(2)>
gap> D := LieDerivedSubalgebra( L );
<Lie algebra of dimension 9 over GF(2)>
gap> LC := LieCentre( L );
<Lie algebra of dimension 7 over GF(2)>
gap> LieLowerNilpotencyIndex( KG );
5
gap> LieUpperNilpotencyIndex( KG );
5
gap> IsLieAbelian( L );
false
gap> IsLieSolvable( L );
true
gap> IsLieMetabelian( L );
false
gap> IsLieCentreByMetabelian( L );
true
]]>
</Example>

</Chapter>