<!-- $Id: funct.xml,v 1.29 2007/02/07 17:35:46 alexk Exp $ -->
<Chapter Label="Funct">
<Heading>&LAGUNA; functions</Heading>
<Section Label="GenSec">
<Heading>General functions for group algebras</Heading>
<ManSection>
<Prop Name="IsGroupAlgebra"
Arg="KG"
Comm="determines if a group ring is a group algebra" />
<Description>
A group ring over a field is called a group algebra.
For a group ring <A>KG</A>,
<C>IsGroupAlgebra</C> returns <K>true</K>,
if the underlying ring of <A>KG</A> is a field;
<K>false</K> is returned otherwise.
This property will be set automatically for every
group ring created by the function <C>GroupRing</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsGroupAlgebra( GroupRing( GF( 2 ), DihedralGroup( 16 ) ) );
true
gap> IsGroupAlgebra( GroupRing( Integers, DihedralGroup( 16 ) ) );
false
]]>
</Example>
<ManSection>
<Prop Name="IsFModularGroupAlgebra"
Arg="KG"
Comm="determines if group algebra is modular" />
<Description>
<Index>modular group algebra</Index>
A group algebra <M>KG</M> over a field <M>K</M> is
called <E>modular</E>, if the characteristic of the field
<M>K</M> divides the order of some element in <M>G</M>.
For a group algebra <A>KG</A> of a finite group
<M>G</M>, <C>IsModularGroupAlgebra</C> returns
<K>true</K>, if <A>KG</A> is modular
according to this definition;
<K>false</K> is returned otherwise.
This property will be set automatically for every
group algebra, created by the function
<C>GroupRing</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsFModularGroupAlgebra( GroupRing( GF( 2 ), SymmetricGroup( 6 ) ) );
true
gap> IsFModularGroupAlgebra( GroupRing( GF( 2 ), CyclicGroup( 3 ) ) );
false
]]>
</Example>
<ManSection>
<Prop Name="IsPModularGroupAlgebra"
Arg="KG"
Comm="determines if group algebra is p-modular" />
<Description>
A group algebra <M>KG</M> is said to be <M>p</M>-modular,
if <M>K</M> is a field of characteristic <M>p</M> and
<M>G</M> is a finite <M>p</M>-group for the same prime
<M>p</M>.
For a group algebra <A>KG</A> of a finite
group <M>G</M>, <C>IsPModularGroupAlgebra</C>
returns <K>true</K>, if <A>KG</A> is
<M>p</M>-modular according to this definition;
<K>false</K> is returned otherwise.
This property will be set automatically for every
group algebra, created by the function
<C>GroupRing</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsPModularGroupAlgebra( GroupRing( GF( 2 ), DihedralGroup( 16 ) ) );
true
gap> IsPModularGroupAlgebra( GroupRing( GF( 2 ), SymmetricGroup( 6 ) ) );
false
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingGroup"
Label="of a group ring"
Arg="KG" />
<Returns>
the underlying group of a group ring
</Returns>
<Description>
This attribute stores the underlying group of a
group ring <A>KG</A>. In fact, it refers to
the attribute <C>UnderlyingMagma</C> which
returns the same result, and was introduced for group
rings for convenience, and for teaching purposes.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF ( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> G := UnderlyingGroup( KG );
<pc group of size 16 with 4 generators>
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingRing"
Arg="KG" />
<Returns>
the underlying ring of a group ring
</Returns>
<Description>
This attribute stores the underlying ring of a
group ring <A>KG</A>. In fact, it refers to
the attribute <C>LeftActingDomain</C> which
returns the same result, and was introduced for group
rings for convenience, and for teaching purposes.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> UnderlyingRing( KG );
GF(2)
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingField"
Arg="KG" />
<Returns>
the underlying field of a group algebra
</Returns>
<Description>
This attribute stores the underlying field of a
group algebra <A>KG</A>. In fact, it refers to
the attribute <C>LeftActingDomain</C> which
returns the same result, and was introduced for group
algebras for convenience, and for teaching purposes.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> UnderlyingField( KG );
GF(2)
]]>
</Example>
</Section>
<!-- ######################################################### -->
<Section Label="ElemFunctions">
<Heading>Operations with group algebra elements</Heading>
<ManSection>
<Attr Name="Support"
Arg="x" />
<Returns>
support of x as a list of elements of the underlying group
</Returns>
<Description>
Returns the support of a group ring element <A>x</A>.
The support of a non-zero element
<M> x = \alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k</M>
of a group ring is the list of elements <M>g_i \in G</M> for which
the coefficient <M>\alpha_i</M> is non-zero. The support of the zero
element of a group ring is defined to be the empty list.
This method is also applicable to elements of magma rings.
</Description>
</ManSection>
<Example>
<![CDATA[
# First we create an element x to use in in the series of examples.
# We map the minimal generating system of the group G to its group algebra
# and denote their images as a and b
gap> G:=DihedralGroup(16);; KG:=GroupRing(GF(2),G);;
gap> l := List( MinimalGeneratingSet( G ), g -> g^Embedding( G, KG ) );
[ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
gap> a := l[1]; b := l[2]; e := One( KG ); # we denote the identity by e
(Z(2)^0)*f1
(Z(2)^0)*f2
(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
gap> Support( x );
[ <identity> of ..., f1, f2, f1*f2 ]
]]>
</Example>
<ManSection>
<Attr Name="CoefficientsBySupport"
Arg="x" />
<Returns>
coefficients of support elements as list of elements
of the underlying ring
</Returns>
<Description>
Returns a list that contains the coefficients
corresponding to the elements of <C>Support( x )</C>
in the same order as the elements appear in
<C>Support( x )</C>.
This method is also applicable to elements of magma rings.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> CoefficientsBySupport( x );
[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ]
]]>
</Example>
<ManSection>
<Attr Name="TraceOfMagmaRingElement"
Arg="x"
Comm="returns the trace of a magma ring element" />
<Returns>
an element of the underlying ring
</Returns>
<Description>
Returns the trace of a group ring element
<A>x</A>.
By definition, the trace of an element
<M> x = \alpha_1 \cdot 1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k </M>
is equal to <M>\alpha_1</M>, that is, the coefficient of the
identity element in <M>G</M>. The trace of the zero
element is zero. This method is also applicable to
elements of magma rings.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> TraceOfMagmaRingElement( x );
Z(2)^0
]]>
</Example>
<ManSection>
<Attr Name="Length"
Arg="x"
Comm="returns the length of a group ring element" />
<Description>
The length of an element of a group ring
<A>x</A> is defined as the number of elements in
its support.
This method is also applicable to elements of magma rings.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> Length( x );
4
]]>
</Example>
<ManSection>
<Attr Name="Augmentation"
Arg="x"
Comm="returns the augmentation of a group ring element" />
<Returns>
the sum of coefficients of a group ring element
</Returns>
<Description>
The augmentation of a group ring element
<M> x = \alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k</M>
is the sum of its coefficients <M> \alpha_1 + \alpha_2 + \cdots + \alpha_k </M>.
The method is also applicable to elements of magma rings.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> Augmentation( x );
0*Z(2)
]]>
</Example>
<ManSection>
<Oper Name="PartialAugmentations"
Arg="KG x"
Comm="returns partial augmentations of a group ring element" />
<Returns>
a list of partial augmentations and a list of conjugacy class
representatives
</Returns>
<Description>
<Index>partial augmentation</Index>
The partial augmentation of an element
<M> x = \alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots +
\alpha_k \cdot g_k</M>
of the group ring <M>KG</M>,
corresponding to the conjugacy class of an element <M>g</M> from
the underlying group <M>G</M> is the sum of coefficients
<M>\alpha_i</M> taken over all <M>g_i</M> such that <M>g_i</M>
is conjugated to <M>g</M>.
The function returns a list of two lists, the first one is a
list of partial augmentations, and the second is a list of
representatives of appropriate conjugacy classes of elements
of the group <M>G</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> y := x + a*b^2;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2+(Z(2)^
0)*f1*f3
gap> PartialAugmentations( KG, y );
[ [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0 ], [ <identity> of ..., f1, f2, f1*f2 ] ]
]]>
</Example>
<ManSection>
<Oper Name="Involution"
Arg="x f"
Comm="Involution determined by mapping" />
<Oper Name="Involution"
Arg="x"
Comm="Classical involution" />
<Returns>
an element of a group ring
</Returns>
<Description>
Let <M>KG</M> be a group ring and let <M>f</M> be a
mapping <M> G \rightarrow G </M>, such that <M>f^2</M>
is the identity mapping on <M>G</M>. Then the involution
of <M>KG</M> induced by <M>f</M> is defined by
<M> \alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k
\mapsto
\alpha_1 \cdot f(g_1) + \alpha_2 \cdot f(g_2) + \cdots + \alpha_k \cdot f(g_k)</M>.
This method returns the image of <A>x</A> under the
involution of <M>KG</M> with respect to <A>f</A>.
<P/>
In the second form the function returns the result of the so-called
classical involution, which is the involution induced by the
map <M> x \mapsto x^{-1}</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> Involution( x );
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f1*f2+(Z(2)^0)*f2*f3*f4
# let's check the action of involution on elements from the group G
gap> l := List( MinimalGeneratingSet( G ), g -> g^Embedding( G, KG ) );
[ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
gap> List( l, Involution );
[ (Z(2)^0)*f1, (Z(2)^0)*f2*f3*f4 ]
gap> List( l, g -> g^-1 );
[ (Z(2)^0)*f1, (Z(2)^0)*f2*f3*f4 ]
]]>
</Example>
<ManSection>
<Attr Name="IsSymmetric"
Arg="x"
Comm="checks whether an element is fixed under the classical involution" />
<Description>
<Index>symmetric element</Index>
An element of a group ring is called <E>symmetric</E> if it is fixed under the
classical involution. This property is checked here.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsSymmetric( x );
false
gap> IsSymmetric( x * Involution( x ) );
true
]]>
</Example>
<ManSection>
<Attr Name="IsUnitary"
Arg="x"
Comm="checks whether an element is unitary w.r.t. classical involution" />
<Description>
<Index>unitary element</Index>
A unit of a group ring is called unitary if the classical involution
inverts it. This property is checked here.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsUnitary(x);
false
# let's check that elements of the group G are unitary
gap> l:=List(MinimalGeneratingSet(G),g -> g^Embedding(G,KG));
[ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
gap> List(l,IsUnitary);
[ true, true ]
]]>
</Example>
<ManSection>
<Meth Name="IsUnit"
Arg="KG x"
Comm="for an element of the specified modular group algebra" />
<Meth Name="IsUnit"
Arg="x"
Comm="for an element without specified magma ring" />
<Description>
This method improves a standard &GAP; functionality for modular group algebras.
<P/>
In the first form the method returns <K>true</K> if
<A>x</A> is an invertible element of the modular
group algebra <A>KG</A> and <K>false</K>
otherwise. This can be done very quickly by checking
whether the augmentation of the element <A>x</A>
is non-zero.
<P/>
In the second form &LAGUNA; first constructs the
group <M>H</M> generated by the
support of <A>x</A>, and, if this group is
a finite <M>p</M>-group, then checks whether
the coefficients of <A>x</A> belong to a
field <M>F</M> of characteristic <M>p</M>. If this is the
case, then <C>IsUnit( FH, x )</C> is called;
otherwise, standard &GAP; method is used.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> x;
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> IsUnit( KG, x ); # clearly, is not a unit due to augmentation zero
false
gap> y := One( KG ) + x; # this should give a unit
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> IsUnit( KG, y );
true
]]>
</Example>
<ManSection>
<Meth Name="InverseOp"
Arg="x"
Comm="for an element of the modular group algebra" />
<Returns>
the inverse element of an element of a group ring
</Returns>
<Description>
This method improves a standard &GAP; functionality for modular group algebras.
It calculates the inverse of a group algebra element.
The user can also invoke this function by typing
<C> x&circum;-1 </C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> y;
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
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>
<ManSection>
<Oper Name="BicyclicUnitOfType1"
Arg="KG a g"
Comm="for elements of a group not embedded into group ring" />
<Oper Name="BicyclicUnitOfType1"
Arg="a g"
Comm="for elements of a group embedded into group ring" />
<Oper Name="BicyclicUnitOfType2"
Arg="KG a g"
Comm="for elements of a group not embedded into group ring" />
<Oper Name="BicyclicUnitOfType2"
Arg="a g"
Comm="for elements of a group embedded into group ring" />
<Returns>
an element of a group ring
</Returns>
<Description>
<Index>bicyclic unit</Index>
let <M>a</M> be an element of order <M>n</M> of a group
<M>G</M>. We put
<M>\alpha = 1 + a + a^2 + ... +a^{n-1} </M>.
Then <M>(a-1)*g*\alpha</M> and <M>\alpha*g*(a-1)</M>
are nilpotent of index two for any element <M>g</M> of the
group <M>G</M> not containing in the normalizer
<M>N_G(\langle a \rangle)</M>, and the units
<M>u_{a,g} = 1 + (a-1) * g * \alpha </M>
and
<M>v_{a,g} = 1 + \alpha * g * (a-1) </M>
are called bicyclic units. Note that
<M>u_{a,g}</M> and <M>v_{a,g}</M> may coincide for some
<M>a</M> and <M>g</M>, but in general this does not hold.
In the three-argument version these methods construct
bicyclic units of both types when <A>a</A> and <A>g</A> are elements
of the underlying group <M>G</M> of a group ring <A>KG</A>.
The two-argument version accepts images of elements <A>a</A> and <A>g</A>
from the underlying group in the group ring <M>KG</M> obtained using the
mapping <C>Embedding( G, KG )</C>.
Note that it is not actually
checked that <M>g</M> is not contained in
<M>N_G(\langle a \rangle)</M>, because this is
verified in <Ref Attr="BicyclicUnitGroup" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup(32,6);
<pc group of size 32 with 5 generators>
gap> KG := GroupRing( GF(2), G );
<algebra-with-one over GF(2), with 5 generators>
gap> g := MinimalGeneratingSet( G );
[ f1, f2 ]
gap> g[1] in Normalizer( G, Subgroup( G, [g[2]] ) );
false
gap> g[2] in Normalizer( G, Subgroup( G, [g[1]] ) );
false
gap> g := List( g, x -> x^Embedding( G, KG ) );
[ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
gap> BicyclicUnitOfType1(g[1],g[2]) = BicyclicUnitOfType2(g[1],g[2]);
false
]]>
</Example>
</Section>
<!-- ######################################################### -->
<Section Label="Ideals">
<Heading>Important attributes of group algebras</Heading>
<ManSection>
<Attr Name="AugmentationHomomorphism"
Arg="KG"
Comm="returns the augmentation homomorphism" />
<Returns>
a homomorphism from a group ring to the underlying ring
</Returns>
<Description>
The mapping which maps an element of a group ring <M>KG</M>
to its augmentation is a homomorphism from <M>KG</M> onto the
ring <M>K</M>; see <Ref Attr="Augmentation" />.
This attribute stores this homomorphism for the group ring
<A>KG</A>.
<P/>
Please note that for calculation of the augmentation of an element
of a group ring the user is strongly recommended to use
<Ref Attr="Augmentation" /> which works much faster than
<C>AugmentationHomomorphism</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := GF( 2 ); G := SymmetricGroup( 3 ); FG := GroupRing( F, G );
GF(2)
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> e := Embedding( G,FG );
<mapping: SymmetricGroup( [ 1 .. 3 ] ) -> AlgebraWithOne( GF(2), ... ) >
gap> x := (1,2)^e; y := (1,3)^e;
(Z(2)^0)*(1,2)
(Z(2)^0)*(1,3)
gap> a := AugmentationHomomorphism( FG );
[ (Z(2)^0)*(1,2,3), (Z(2)^0)*(1,2) ] -> [ Z(2)^0, Z(2)^0 ]
gap> x^a; y^a; ( x + y )^a; # this is slower
Z(2)^0
Z(2)^0
0*Z(2)
gap> Augmentation(x); Augmentation(y); Augmentation( x + y ); # this is faster
Z(2)^0
Z(2)^0
0*Z(2)
]]>
</Example>
<ManSection>
<Attr Name="AugmentationIdeal"
Arg="KG"
Comm="returns the augmentation ideal" />
<Returns>
an ideal of a group ring
</Returns>
<Description>
If <M>KG</M> is a group ring, then its augmentation
ideal <M>A</M> is generated by all elements of the form
<M>g-1</M>, where <M>g \in G</M> &bslash; &obrace; <M>1</M> &cbrace;.
The augmentation ideal
consists of all elements of <M>FG</M> with augmentation
<M>0</M>; see <Ref Meth="Augmentation" />.
This method changes a standard &GAP; functionality for modular group algebras and returns the augmentation
ideal of a modular group algebra <A>KG</A>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> AugmentationIdeal( KG );
<two-sided ideal in <algebra-with-one over GF(2), with 4 generators>,
(dimension 15)>
]]>
</Example>
<ManSection>
<Attr Name="RadicalOfAlgebra"
Arg="KG"
Comm="for modular group algebra of finite p-group" />
<Returns>
an ideal of a group algebra
</Returns>
<Description>
This method improves a standard &GAP; functionality for modular group algebras of finite <M>p</M>-groups.
Since in this case the radical of the group algebra coincides with
its augmentation ideal, this method simply checks if
the algebra <A>KG</A> is a <M>p</M>-modular group algebra,
and, if yes, it returns the augmentation ideal;
otherwise, the standard &GAP; method will be used.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> RadicalOfAlgebra( KG );
<two-sided ideal in <algebra-with-one over GF(2), with 4 generators>,
(dimension 15)>
gap> RadicalOfAlgebra( KG ) = AugmentationIdeal( KG );
true
]]>
</Example>
<ManSection>
<Attr Name="WeightedBasis"
Arg="KG"
Comm="for modular group algebra of finite p-group" />
<Returns>
a record of two components: weighted basis elements and
their weights
</Returns>
<Description>
The argument <A>KG</A> must be a <M>p</M>-modular
group algebra.
<P/>
For a group algebra <M>KG</M>, let <M>A</M> denote the
augmentation ideal, and assume that <M>c</M> is the
smallest number such that <M>A^c=0</M>. Then a weighted
basis of <M>KG</M> is some basis <M> b_1, \ldots, b_n </M>
for the augmentation ideal <M>A</M>, for which there are
indices <M> i_1=1, \ldots, i_{c-1} </M> such that
<M> b_{i_k}, \ldots, b_n </M> is a basis for <M>A^k</M>.
The weight of an element <M>b_i</M> of a weighted basis
is the unique integer <M>w</M> such that
<M>b_i</M> belongs to <M>w</M>-th power of <M>A</M> but
does not belong to its <M>(w+1)</M>-th power.
<P/>
Note that this function actually constructs a basis for
the <E>augmentation ideal</E> of <A>KG</A> and not for
<A>KG</A> itself. Since the augmentation ideal has
co-dimension 1 in <C>KG</C>, a basis for <C>KG</C> can
be easily obtained by adjoining the identity element of
the group.
<P/>
The method returns a record whose basis entry is the basis
and the weights entry is a list of the corresponding weights
the of basis elements. See Section <Ref Sect="TheoryThird" />
for more details.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), ElementaryAbelianGroup( 4 ) );
<algebra-with-one over GF(2), with 2 generators>
gap> WeightedBasis( KG );
rec(
weightedBasis := [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2, (Z(2)^0)*<identity>
of ...+(Z(2)^0)*f1, (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(
Z(2)^0)*f1*f2 ], weights := [ 1, 1, 2 ] )
]]>
</Example>
<ManSection>
<Attr Name="AugmentationIdealPowerSeries"
Arg="KG"
Comm="for p-modular group algebra" />
<Returns>
a list of ideals of a group algebra
</Returns>
<Description>
The argument <A>KG</A> is a <M>p</M>-modular group
algebra. The method returns a list whose elements are
the terms of the augmentation ideal filtration of
<A>KG</A>, that is
<C>AugmentationIdealPowerSeries(A)[i]</C> is the
<M>i</M>-th power of the augmentation ideal of <A>KG</A>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
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)> ]
gap> Length(last);
9
]]>
</Example>
<ManSection>
<Attr Name="AugmentationIdealNilpotencyIndex"
Arg="KG"
Comm="for p-modular group algebra" />
<Description>
For the <M>p</M>-modular group algebra <A>KG</A>
the method returns the smallest number <M>n</M> such
that <M>A^n=0</M>, where <M>A</M> is the
augmentation ideal of <A>KG</A>. This can be
done using Jenning's theory without the explicit
calculations of the powers of the augmentation ideal.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> AugmentationIdealNilpotencyIndex( KG );
9
]]>
</Example>
<ManSection>
<Attr Name="AugmentationIdealOfDerivedSubgroupNilpotencyIndex"
Arg="KG"
Comm="for p-modular group algebra" />
<Description>
For the <M>p</M>-modular group algebra <A>KG</A>
this attribute stores the nilpotency index of the
augmentation ideal of <M>KG'</M> where <M>G'</M> denotes the
derived subgroup of <M>G</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> AugmentationIdealOfDerivedSubgroupNilpotencyIndex( KG );
4
gap> D := DerivedSubgroup( UnderlyingGroup( KG ) );
Group([ f3, f4 ])
gap> KD := GroupRing( GF( 2 ), D );
<algebra-with-one over GF(2), with 2 generators>
gap> AugmentationIdealNilpotencyIndex( KD );
4
]]>
</Example>
<ManSection>
<Oper Name="LeftIdealBySubgroup"
Arg="KG H"
Comm="returns the left or twosided ideal of a group ring" />
<Oper Name="RightIdealBySubgroup"
Arg="KG H"
Comm="returns the right or twosided ideal of a group ring" />
<Oper Name="TwoSidedIdalBySubgroup"
Arg="KG H"
Comm="returns the twosided ideal of a group ring" />
<Oper Name="LeftIdealBySubgroup"
Arg="KG H"
Comm="returns the twosided ideal of a group ring" />
<Returns>
an ideal of a group ring
</Returns>
<Description>
Let <A>KG</A> be a group ring of a group <M>G</M> over the
ring <M>K</M>, and <A>H</A> be a subgroup of <M>G</M>.
Then the set <M>J_l(H)</M> of all elements of <A>KG</A>
of the form
<Display>
\sum_{h \in H} x_h(h-1)
</Display>
is the left ideal in <A>KG</A> generated by all elements
<M>h-1</M> with <M>h</M> in <M>H</M>. The right ideal
<M>J_r(H)</M> is defined analogously.
These operations are used to consrtuct such ideals,
taking into account the fact, that the ideal <M>J_l(H)</M>
is two-sided if and only if <A>H</A> is normal in <M>G</M>.
An attempt of constructing two-sided ideal for a non-normal
subgroup <A>H</A> will lead to an error message.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF(2), DihedralGroup(16) );
<algebra-with-one over GF(2), with 4 generators>
gap> G := DihedralGroup(16);
<pc group of size 16 with 4 generators>
gap> KG := GroupRing( GF(2), G );
<algebra-with-one over GF(2), with 4 generators>
gap> D := DerivedSubgroup( G );
Group([ f3, f4 ])
gap> LeftIdealBySubgroup( KG, D );
<two-sided ideal in <algebra-with-one over GF(2), with 4 generators>,
(dimension 12)>
gap> H := Subgroup( G, [ GeneratorsOfGroup(G)[1] ]);
Group([ f1 ])
gap> IsNormal( G, H );
false
gap> LeftIdealBySubgroup( KG, H );
<left ideal in <algebra-with-one over GF(2), with 4 generators>,
(dimension 8)>
]]>
</Example>
</Section>
<!-- ######################################################### -->
<Section Label="UnitGroup">
<Heading>Computations with the unit group</Heading>
<ManSection>
<Attr Name="NormalizedUnitGroup"
Arg="KG" />
<Returns>
a group generated by group algebra elements
</Returns>
<Description>
Determines the normalized unit group of a <M>p</M>-modular
group algebra <A>KG</A> over the field of <M>p</M> elements.
Returns the normalized unit group
as the group generated by certain elements of <A>KG</A>;
see Section <Ref Sect="TheoryThird" /> for more details.
<P/>
For efficient computations the user is recommended to use
<Ref Attr="PcNormalizedUnitGroup" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> V := NormalizedUnitGroup( KG );
<group of size 32768 with 15 generators>
gap> u := GeneratorsOfGroup( V )[4];
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
]]>
</Example>
<ManSection>
<Attr Name="PcNormalizedUnitGroup"
Arg="KG" />
<Returns>
a group given by power-commutator presentation
</Returns>
<Description>
The argument <A>KG</A> is a <M>p</M>-modular group
algebra over the field of <M>p</M> elements.
<C>PcNormalizedUnitGroup</C> returns
the normalized unit group of <A>KG</A> given by a power-commutator
presentation. The generators in this polycyclic presentation
correspond to the weighted basis elements of <A>KG</A>.
For more details, see Section <Ref Sect="TheoryThird" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> W := PcNormalizedUnitGroup( KG );
<pc group of size 32768 with 15 generators>
gap> w := GeneratorsOfGroup( W )[4];
f4
]]>
</Example>
<ManSection>
<Attr Name="NaturalBijectionToPcNormalizedUnitGroup"
Arg="KG" />
<Returns>
a homomorphism of groups
</Returns>
<Description>
The normalised unit group of a <M>p</M>-modular group
algebra <M>KG</M> over the field of <M>p</M> elements
can be computed using two methods,
namely <Ref Attr="NormalizedUnitGroup" /> and
<Ref Attr="PcNormalizedUnitGroup" />.
These two methods return two different objects, and they
can be used for different types of computations. The
elements of <C>NormalizedUnitGroup(KG)</C> are
represented in their natural group algebra representation,
and hence they can easily be identified in the group
algebra. However, the more quickly constructed
<C>NormalizedUnitGroup(KG)</C> is often not suitable
for further fast calculations. Hence one will have to use
<C>PcNormalizedUnitGroup(KG)</C> if one wants to
find some group theoretic properties of the normalized
unit group. This method returns the bijection from
<C>NormalizedUnitGroup(<A>KG</A>)</C> onto
<C>PcNormalizedUnitGroup(<A>KG</A>)</C>. This bijection can
be used to map the result of a computation in
<C>PcNormalizedUnitGroup(<A>KG</A>)</C> into <C>NormalizedUnitGroup(<A>KG</A>)</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> f := NaturalBijectionToPcNormalizedUnitGroup( KG );
MappingByFunction( <group of size 32768 with
15 generators>, <pc group of size 32768 with
15 generators>, function( x ) ... end )
gap> u := GeneratorsOfGroup( V )[4];;
gap> u^f;
f4
gap> GeneratorsOfGroup( V )[4]^f = GeneratorsOfGroup( W )[4];
true
]]>
</Example>
<ManSection>
<Attr Name="NaturalBijectionToNormalizedUnitGroup"
Arg="KG" />
<Returns>
a homomorphism of groups
</Returns>
<Description>
For a <M>p</M>-modular group algebra <A>KG</A>
over the field of <M>p</M> elements
this function returns the inverse of the mapping
<Ref Attr="NaturalBijectionToPcNormalizedUnitGroup" />
</Description>
</ManSection>
<Example>
<![CDATA[
gap> t := NaturalBijectionToNormalizedUnitGroup(KG);;
gap> w := GeneratorsOfGroup(W)[4];;
gap> w^t;
(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
gap> GeneratorsOfGroup( W )[4]^t = GeneratorsOfGroup( V )[4];
true
]]>
</Example>
<ManSection>
<Oper Name="Embedding"
Arg="H V" />
<Returns>
a homomorphism from an underlying group to a normalized unit group in pc-presentation
</Returns>
<Description>
Let <A>H</A> be a subgroup of a group <M>G</M> and
<A>V</A> be the normalized unit group of the group algebra
<M>KG</M> given by the power-commutator presentation
(see <Ref Attr="PcNormalizedUnitGroup"/>.
Then <C>Embedding( H, V )</C> returns the homomorphism from
<A>H</A> to <A>V</A>, which is the composition of
<C>Embedding( H, KG )</C> and
<C>NaturalBijectionToPcNormalizedUnitGroup( KG )</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>
gap> KG := GroupRing( GF( 2 ), G );
<algebra-with-one over GF(2), with 4 generators>
gap> V:=PcNormalizedUnitGroup( KG );
<pc group of size 32768 with 15 generators>
gap> ucs := UpperCentralSeries( V );
[ <pc group of size 32768 with 15 generators>,
<pc group of size 4096 with 12 generators>,
Group([ f3*f5*f13*f15, f7, f15, f13*f15, f14*f15, f11*f13*f14*f15, f12,
f9*f12, f10 ]),
Group([ f3*f5*f13*f15, f7, f15, f13*f15, f14*f15, f11*f13*f14*f15 ]),
Group([ ]) ]
gap> f := Embedding( G, V );
[ f1, f2, f3, f4 ] -> [ f2, f1, f3, f7 ]
gap> G1 := Image( f, G );
Group([ f2, f1, f3, f7 ])
gap> H := Intersection( ucs[2], G1 );
Group([ f3, f7, f3*f7 ])
# H is the intersection of G and the 3rd centre of V(KG)
gap> T:=PreImage( f, H );
Group([ f3, f4, f3*f4 ])
# and T is its preimage in G
gap> IdGroup( T );
[ 4, 1 ]
]]>
</Example>
<ManSection>
<Attr Name="Units"
Arg="KG" />
<Returns>
the unit group of a group ring
</Returns>
<Description>
This improves a standard &GAP; functionality for modular group
algebras of finite <M>p</M>-groups over the field of <M>p</M> elements.
It returns the unit group of <A>KG</A> as a direct product of
<C>Units(K)</C> and <C>NormalizedUnitGroup(KG)</C>, where the latter is
generated by certain elements of <A>KG</A>;
see Chapter <Ref Chap="Theory" /> for more details.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> U := Units( KG );
#I LAGUNA package: Computing the unit group ...
<group of size 32768 with 15 generators>
# now elements of U are already in KG
gap> GeneratorsOfGroup( U )[5];
(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f2*f3
# in the next example the direct product structure is more clear
gap> FH := GroupRing( GF(3), SmallGroup(27,3) );
<algebra-with-one over GF(3), with 3 generators>
gap> T := Units( FH );
#I LAGUNA package: Computing the unit group ...
<group of size 5083731656658 with 27 generators>
gap> x := GeneratorsOfGroup( T )[1];
Tuple( [ Z(3), (Z(3)^0)*<identity> of ... ] )
gap> x in FH;
false
gap> x[1] * x[2] in FH;
true # this is the way to get the corresponding element of FH
]]>
</Example>
<ManSection>
<Attr Name="PcUnits"
Arg="KG" />
<Returns>
a group given by power-commutator presentation
</Returns>
<Description>
Returns the unit group of <A>KG</A> as a direct product of
<C>Units(K)</C> and <C>PcNormalizedUnitGroup(KG)</C>, where the latter
is a group given by a polycyclic presentation.
See Section <Ref Sect="TheoryFourth" /> for more details.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> W := PcUnits( KG );
<pc group of size 32768 with 15 generators>
gap> GeneratorsOfGroup( W )[5];
f5
# in the next example the direct product structure is more clear
gap> FH := GroupRing( GF(3), SmallGroup(27,3) );
<algebra-with-one over GF(3), with 3 generators>
gap> T := PcUnits(FH);
<group of size 5083731656658 with 27 generators>
gap> x := GeneratorsOfGroup( T )[2];
Tuple( [ Z(3)^0, f1 ] )
]]>
</Example>
<ManSection>
<Prop Name="IsGroupOfUnitsOfMagmaRing"
Arg="U" />
<Description>
This property will be automatically set
<K>true</K>, if <A>U</A> is a group generated by
some units of a magma ring, including
<C>Units(KG)</C> and <C>NormalizedUnitgroup(KG)</C>.
Otherwise this property will not be bound.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsGroupOfUnitsOfMagmaRing( NormalizedUnitGroup( KG ) );
true
gap> IsGroupOfUnitsOfMagmaRing( Units( KG ) );
true
]]>
</Example>
<ManSection>
<Prop Name="IsUnitGroupOfGroupRing"
Arg="U"/>
<Description>
This property will be automatically set <K>true</K>,
if <A>U</A> is the unit group of a <M>p</M>-modular group
algebra, obtained either by <C>Units(KG)</C> or by
<C>PcUnits(KG)</C>. Otherwise this property will not be
bound.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsUnitGroupOfGroupRing( Units( KG ) );
true
gap> IsUnitGroupOfGroupRing( PcUnits( KG ) );
true
]]>
</Example>
<ManSection>
<Prop Name="IsNormalizedUnitGroupOfGroupRing"
Arg="U" />
<Description>
This property will be automatically set <K>true</K>,
if <A>U</A> is the normalized unit group of a
<M>p</M>-modular group algebra, obtained either by
<C>NormalizedUnitGroup(KG)</C> or by
<C>PcNormalizedUnitGroup(KG)</C>.
Otherwise this property will not be bound.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsNormalizedUnitGroupOfGroupRing( NormalizedUnitGroup( KG ) );
true
gap> IsNormalizedUnitGroupOfGroupRing( PcNormalizedUnitGroup( KG ) );
true
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingGroupRing"
Arg="U" />
<Returns>
a group ring
</Returns>
<Description>
If <A>U</A> is the (normalized) unit group of a
<M>p</M>-modular group algebra <M>KG</M>
obtained using one of the
functions <C>Units(KG)</C>, <C>PcUnits(KG)</C>,
<C>NormalizedUnitGroup(KG)</C> or
<C>PcNormalizedUnitGroup(KG)</C>,
then the attribute <C>UnderlyingGroupRing</C> stores
<M>KG</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> UnderlyingGroupRing( Units( KG ) );
<algebra-with-one over GF(2), with 4 generators>
gap> UnderlyingGroupRing( PcUnits( KG ) );
<algebra-with-one over GF(2), with 4 generators>
gap> UnderlyingGroupRing( NormalizedUnitGroup( KG ) );
<algebra-with-one over GF(2), with 4 generators>
gap> UnderlyingGroupRing( PcNormalizedUnitGroup( KG ) );
<algebra-with-one over GF(2), with 4 generators>
]]>
</Example>
<ManSection>
<Attr Name="UnitarySubgroup"
Arg="U" />
<Returns>
the subgroup of the unit group
</Returns>
<Description>
Let <A>U</A> be the normalized unit group of a group ring
in either natural (see <Ref Attr="NormalizedUnitGroup"/>)
or power-commutator (see <Ref Attr="PcNormalizedUnitGroup"/>)
presentation.
The attribute stores the unitary subgroup of <A>U</A>,
generated by all unitary units of <A>U</A>
(see <Ref Attr="IsUnitary"/>).
The method is straightforward, so it is not recommended
to run it for large groups.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 8 ) );
<algebra-with-one over GF(2), with 3 generators>
gap> U := NormalizedUnitGroup( KG );
<group of size 128 with 7 generators>
gap> HU := UnitarySubgroup( U );
<group with 5 generators>
gap> IdGroup( HU );
[ 64, 261 ]
gap> V := PcNormalizedUnitGroup( KG );
<pc group of size 128 with 7 generators>
gap> HV := UnitarySubgroup( V );
Group([ f1, f2, f5, f6, f7 ])
gap> IdGroup( HV );
[ 64, 261 ]
gap> Image(NaturalBijectionToPcNormalizedUnitGroup( KG ), HU ) = HV;
true
]]>
</Example>
<ManSection>
<Attr Name="BicyclicUnitGroup"
Arg="U" />
<Returns>
the subgroup of the unit group, generated by bicyclic units
</Returns>
<Description>
Let <A>U</A> be the normalized unit group of a group ring
in either natural (see <Ref Attr="NormalizedUnitGroup"/>)
or power-commutator (see <Ref Attr="PcNormalizedUnitGroup"/>)
presentation.
The attribute stores the subgroup of <A>U</A>,
generated by all bicyclic units <M>u_{g,h}</M> and
<M>v_{g,h}</M> (see <Ref Oper="BicyclicUnitOfType1"/> and
<Ref Oper="BicyclicUnitOfType1"/>),
where <M>g</M> and <M>h</M> run over the elements of the
underlying group, and <M>h</M> do not belongs to the
normalizer of <M> \langle g \rangle </M> in <M>G</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 8 ) );
<algebra-with-one over GF(2), with 3 generators>
gap> U := NormalizedUnitGroup( KG );
<group of size 128 with 7 generators>
gap> BU := BicyclicUnitGroup( U );
<group with 2 generators>
gap> IdGroup( BU );
[ 4, 2 ]
gap> V := PcNormalizedUnitGroup( KG );
<pc group of size 128 with 7 generators>
gap> BV := BicyclicUnitGroup( V );
Group([ f5*f6, f6*f7 ])
gap> IdGroup( BV );
[ 4, 2 ]
gap> Image( NaturalBijectionToPcNormalizedUnitGroup( KG ), BU ) = BV;
true
]]>
</Example>
<ManSection>
<Oper Name="AugmentationIdealPowerFactorGroup"
Arg="KG n" />
<Returns>
PcGroup
</Returns>
<Description>
????????????????????????????????????????
</Description>
</ManSection>
<Example>
<![CDATA[
]]>
</Example>
<ManSection>
<Attr Name="GroupBases"
Arg="KG"
Comm="for a p-modular group algebra" />
<Returns>
a list of lists of group rings elements
</Returns>
<Description>
The subgroup <M>B</M> of the normalized unit group of the
group algebra <M>KG</M> is called a <E>group basis</E>, if
the elements of <M>B</M> are linearly independent over the
field <M>K</M> and <M> KB=KG </M>.
If <A>KG</A> is a <M>p</M>-modular group algebra, then
<C>GroupBases</C> returns a list of representatives of
the conjugacy classes of the group bases of the group
algebra <A>KG</A> in its normalised unit group.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> D8 := DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> K := GF(2);
GF(2)
gap> KD8 := GroupRing( GF( 2 ), D8 );
<algebra-with-one over GF(2), with 3 generators>
gap> gb := GroupBases( KD8 );;
gap> Length( gb );
32
gap> gb[1];
[ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f3,
(Z(2)^0)*f1*f2+(Z(2)^0)*f2*f3+(Z(2)^0)*f1*f2*f3,
(Z(2)^0)*f2+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f2*f3,
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f2*f3+(Z(2)^
0)*f1*f2*f3, (Z(2)^0)*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f2*f3,
(Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^
0)*f2*f3, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f2*f3 ]
gap> Length( last );
8
]]>
</Example>
</Section>
<!-- ######################################################### -->
<Section Label="LieAlgebra">
<Heading>The Lie algebra of a group algebra</Heading>
<ManSection>
<Meth Name="LieAlgebraByDomain"
Arg="A" />
<Description>
This method takes a group algebra as its argument,
and constructs its associated Lie algebra in which the product
is the bracket operation: <M>[a,b]=ab-ba</M>.
It is recommended that the user never calls this method.
The Lie algebra for an associative algebra should normally
be created using <C>LieAlgebra( A )</C>.
When <C>LieAlgebra</C> is first invoked, it constructs the
Lie algebra for <A>A</A> using <C>LieAlgebraByDomain</C>.
After that it stores this Lie algebra and simply returns it
if <C>LieAlgebra</C> is called again.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup(3);; FG := GroupRing( GF( 2 ), G );
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
]]>
</Example>
<ManSection>
<Filt Name="IsLieAlgebraByAssociativeAlgebra"
Arg="L"
Type="Category" />
<Description>
This category signifies that the Lie algebra <A>L</A> was
constructed as the Lie algebra associated with an associative
algebra (this piece of information cannot be obtained later).
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF(3), DihedralGroup(16) );
<algebra-with-one over GF(3), with 4 generators>
gap> L := LieAlgebra ( KG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(3)>
gap> IsLieAlgebraByAssociativeAlgebra( L );
true
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingAssociativeAlgebra"
Arg="L" />
<Returns>
the underlying associative algebra of a Lie algebra
</Returns>
<Description>
If a Lie algebra <A>L</A> is constructed from an associative
algebra, then it remembers this underlying associative algebra as
one of its attributes.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF(2), DihedralGroup(16) );
<algebra-with-one over GF(2), with 4 generators>
gap> L := LieAlgebra ( KG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> UnderlyingAssociativeAlgebra( L );
<algebra-with-one over GF(2), with 4 generators>
gap> last = KG;
true
]]>
</Example>
<ManSection>
<Attr Name="NaturalBijectionToLieAlgebra"
Arg="A" />
<Returns>
a mapping
</Returns>
<Description>
The natural linear bijection between the (isomorphic, but not
equal) underlying vector spaces of an associative algebra
<A>A</A> and its associated Lie algebra is stored as an
attribute of <A>A</A>. Note that this is a vector space
isomorphism between two algebras, but not an algebra
isomorphism.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := GF( 2 ); G := SymmetricGroup( 3 ); FG := GroupRing( F, G );
GF(2)
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> t := NaturalBijectionToLieAlgebra( FG );
MappingByFunction( <algebra-with-one over GF(2), with
2 generators>, <Lie algebra over GF(
2)>, <Operation "LieObject">, function( y ) ... end )
gap> a := Random( FG );
(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2)+(Z(2)^0)*(1,3)
gap> a * a; # product in the associative algebra
(Z(2)^0)*()+(Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2)
gap> b := a^t;
LieObject( (Z(2)^0)*(1,2,3)+(Z(2)^0)*(1,3,2)+(Z(2)^0)*(1,3) )
gap> b * b; # product in the Lie algebra (commutator) ...
LieObject( <zero> of ... ) # ... must be zero!
]]>
</Example>
<ManSection>
<Attr Name="NaturalBijectionToAssociativeAlgebra"
Arg="L" />
<Description>
This is the inverse of the previous linear bijection,
stored as an attribute of the Lie algebra <A>L</A>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup(3); FG := GroupRing( GF( 2 ), G );
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> s := NaturalBijectionToAssociativeAlgebra( L );
MappingByFunction( <Lie algebra over GF(2)>, <algebra-with-one over GF(
2), with 2 generators>, function( y ) ... end, <Operation "LieObject"> )
gap> InverseGeneralMapping( s ) = NaturalBijectionToLieAlgebra( FG );
true
]]>
</Example>
<ManSection>
<Prop Name="IsLieAlgebraOfGroupRing"
Arg="L" />
<Description>
If a Lie algebra <A>L</A> is constructed from an associative
algebra which happens to be in fact a group ring, it has
many nice properties that can be used for fast algorithms,
so this information is stored as a property.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := GF( 2 ); G := SymmetricGroup( 3 ); FG := GroupRing( F, G );
GF(2)
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsLieAlgebraOfGroupRing( L );
true
]]>
</Example>
<ManSection>
<Attr Name="UnderlyingGroup"
Label="of Lie algebra of a group ring"
Arg="L" />
<Returns>
the underlying group
</Returns>
<Description>
The underlying group of a Lie algebra <A>L</A> that is
constructed from a group ring is defined as the underlying
group of this group ring; see <Ref Attr="UnderlyingGroup"
Label="of a group ring" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := GF( 2 ); G := SymmetricGroup( 3 ); FG := GroupRing( F, G );
GF(2)
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> UnderlyingGroup( L );
Sym( [ 1 .. 3 ] )
gap> LeftActingDomain( L );
GF(2)
]]>
</Example>
<ManSection>
<Oper Name="Embedding"
Arg="U L" />
<Returns>
a mapping, which is a composition of two mappings
</Returns>
<Description>
Let <M>FG</M> be a group ring, let <A>U</A> be a submagma
of <M>G</M>, and let <A>L</A> be the Lie algebra
associated with <M>FG</M>. Then <C>Embedding(<A>U</A>, <A>L</A> )</C>
returns the obvious mapping from <A>U</A> to <A>L</A>
(as the composition of the mappings
<C>Embedding( <A>U</A>, FG )</C> and
<C>NaturalBijectionToLieAlgebra( FG )</C>).
</Description>
</ManSection>
<Example>
<![CDATA[
gap> F := GF( 2 ); G := SymmetricGroup( 3 ); FG := GroupRing( F, G );
GF(2)
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> f := Embedding( G, L );
CompositionMapping( MappingByFunction( <algebra-with-one over GF(2), with
2 generators>, <Lie algebra over GF(
2)>, <Operation "LieObject">, function( y ) ... end ), <mapping: SymmetricGrou\
p( [ 1 .. 3 ] ) -> AlgebraWithOne( GF(2), ... ) > )
gap> (1,2)^f + (1,3)^f;
LieObject( (Z(2)^0)*(1,2)+(Z(2)^0)*(1,3) )
]]>
</Example>
<ManSection>
<Meth Name="LieCentre"
Arg="L" />
<Returns>
a Lie algebra
</Returns>
<Description>
The centre of the Lie algebra associated with a group ring
corresponds to the centre of the underlying group ring,
and it can be calculated very fast by considering the
conjugacy classes of the group. This method returns the
centre of <A>L</A> using this idea.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup( 256, 400 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 256 with 8 generators>
<algebra-with-one over GF(2), with 8 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> C := LieCentre( L );
<Lie algebra of dimension 28 over GF(2)>
gap> D := LieDerivedSubalgebra( L );
#I LAGUNA package: Computing the Lie derived subalgebra ...
<Lie algebra of dimension 228 over GF(2)>
gap> c := Dimension( C ); d := Dimension( D ); l := Dimension( L );
28
228
256
gap> c + d = l;
true # This is always the case for Lie algebras of group algebras!
]]>
</Example>
<ManSection>
<Meth Name="LieDerivedSubalgebra"
Arg="L" />
<Returns>
a Lie algebra
</Returns>
<Description>
If <A>L</A> is the Lie algebra associated with a group ring,
then this method returns the Lie derived subalgebra of
<A>L</A>. This can be done very fast using the conjugacy classes
of the underlying group.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup( 256, 400 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 256 with 8 generators>
<algebra-with-one over GF(2), with 8 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> C := LieCentre( L );
<Lie algebra of dimension 28 over GF(2)>
gap> D := LieDerivedSubalgebra( L );
#I LAGUNA package: Computing the Lie derived subalgebra ...
<Lie algebra of dimension 228 over GF(2)>
gap> l := Dimension( L ); c := Dimension( C ); d := Dimension( D );
256
28
228
gap> c + d = l;
true # This is always the case for Lie algebras of group algebras!
]]>
</Example>
<ManSection>
<Meth Name="IsLieAbelian"
Arg="L" />
<Description>
The Lie algebra <A>L</A> of an associative algebra <M>A</M>
is Lie abelian, if and only if <M>A</M> is abelian, so this
method refers to <C>IsAbelian( A )</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup( 3 ); FG := GroupRing( GF( 2 ), G);
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsAbelian( G );
false
gap> IsAbelian( L ); # This command should never be used for Lie algebras!
true # It gives a result, but (probably) not the desired one.
gap> IsLieAbelian( L ); # Instead, IsLieAbelian is the correct command.
false
]]>
</Example>
<ManSection>
<Meth Name="IsLieSolvable"
Arg="L" />
<Description>
In <Cite Key="PPS73" /> Passi, Passman, and Sehgal have
classified all groups <M>G</M> such that the Lie algebra associated
with the group ring is solvable.
This method uses their classification, making it
considerably faster than the more elementary method
which just calculates Lie commutators.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup( 256, 400 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 256 with 8 generators>
<algebra-with-one over GF(2), with 8 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsLieSolvable( L ); # This is very fast.
true
gap> List( LieDerivedSeries( L ), Dimension ); # This is very slow.
[ 256, 228, 189, 71, 0 ]
]]>
</Example>
<ManSection>
<Meth Name="IsLieNilpotent"
Arg="L" />
<Description>
In <Cite Key="PPS73" /> Passi, Passman, and Sehgal have classified
all groups <M>G</M> such that the Lie algebra associated
with the group ring is Lie nilpotent. This method
uses their classification, making it considerably faster
than the more elementary method which just calculates
Lie commutators.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup( 256, 400 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 256 with 8 generators>
<algebra-with-one over GF(2), with 8 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsLieNilpotent( L ); # This is very fast.
true
gap> List( LieLowerCentralSeries( L ), Dimension ); # This is very slow.
[ 256, 228, 222, 210, 191, 167, 138, 107, 76, 54, 29, 15, 6, 0 ]
]]>
</Example>
<ManSection>
<Prop Name="IsLieMetabelian"
Arg="L" />
<Description>
In <Cite Key="LR86" /> Levin and Rosenberger have classified
all groups <M>G</M> such that the Lie algebra associated with
the group ring is Lie metabelian. This method uses
their classification, making it considerably faster than
the more elementary method which just calculates Lie
commutators.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SmallGroup( 256, 400 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 256 with 8 generators>
<algebra-with-one over GF(2), with 8 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsLieMetabelian( L );
false
]]>
</Example>
<ManSection>
<Prop Name="IsLieCentreByMetabelian"
Arg="L" />
<Description>
In <Cite Key="Ross" /> the third author of this package
classified all groups <M>G</M> such that the Lie algebra associated
with the group ring is Lie
centre-by-metabelian.
This method uses the classification, making it
considerably faster than the more elementary method
which just calculates Lie commutators.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup( 3 ); FG := GroupRing( GF( 2 ), G );
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> IsLieMetabelian( L );
false
gap> IsLieCentreByMetabelian( L );
true
]]>
</Example>
<ManSection>
<Meth Name="CanonicalBasis"
Arg="L" />
<Returns>
basis of a Lie algebra
</Returns>
<Description>
The canonical basis of a group algebra <M>FG</M> is formed
by the elements of <M>G</M>. Here <A>L</A> is the
Lie algebra associated with <M>FG</M>, and
the method returns the images of the elements of <M>G</M> in
<A>L</A>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup( 3 ); FG := GroupRing( GF( 2 ), G );
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> B := CanonicalBasis( L );
CanonicalBasis( <Lie algebra of dimension 6 over GF(2)> )
gap> Elements( B );
[ LieObject( (Z(2)^0)*() ), LieObject( (Z(2)^0)*(2,3) ),
LieObject( (Z(2)^0)*(1,2) ), LieObject( (Z(2)^0)*(1,2,3) ),
LieObject( (Z(2)^0)*(1,3,2) ), LieObject( (Z(2)^0)*(1,3) ) ]
]]>
</Example>
<ManSection>
<Prop Name="IsBasisOfLieAlgebraOfGroupRing"
Arg="B" />
<Description>
A basis <A>B</A> has this property if the preimages of the basis
vectors in the group algebra form a group. It can be verified if a
basis has this property. This is
important for the speed of the calculation of the structure
constants table; see <Ref Meth="StructureConstantsTable" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := SymmetricGroup( 3 ); FG := GroupRing( GF( 2 ), G );
Sym( [ 1 .. 3 ] )
<algebra-with-one over GF(2), with 2 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> B := CanonicalBasis( L );
CanonicalBasis( <Lie algebra of dimension 6 over GF(2)> )
gap> IsBasisOfLieAlgebraOfGroupRing( B );
true
]]>
</Example>
<ManSection>
<Meth Name="StructureConstantsTable"
Arg="B" />
<Description>
A very fast implementation for calculating the structure
constants table for the Lie algebra <C>L</C> associated
with a group ring with respect to its canonical basis
<A>B</A> using its special structure;
see <Ref Meth="CanonicalBasis" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := CyclicGroup( 2 ); FG := GroupRing( GF( 2 ), G );
<pc group of size 2 with 1 generators>
<algebra-with-one over GF(2), with 1 generators>
gap> L := LieAlgebra( FG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> B := CanonicalBasis( L );
CanonicalBasis( <Lie algebra of dimension 2 over GF(2)> )
gap> StructureConstantsTable( B );
[ [ [ [ ], [ ] ], [ [ ], [ ] ] ], [ [ [ ], [ ] ], [ [ ], [ ] ] ], -1,
0*Z(2) ]
]]>
</Example>
<ManSection>
<Attr Name="LieUpperNilpotencyIndex"
Arg="KG" />
<Description>
<Index>upper Lie power series</Index>
In a modular group algebra <M>KG</M> the <E>upper
Lie power series</E> is defined as follows:
<M>KG^{(1)}=KG</M>, <M>KG^{(n+1)}</M> is the associative
ideal, generated by <M>[KG^{(n)},KG]</M>.
The upper Lie nilpotency index <M>t^L(G)</M> of the group
algebra <M>KG</M> is defined to be the
smallest number <M>n</M> such that <M>KG^{(n)}=0</M>.
It can be calculated very fast using Lie dimension
subgroups <Cite Key="Shalev91" />, that is, using only
information about the underlying group; see
<Ref Attr="LieDimensionSubgroups" />. This is why it is
stored as an attribute of the group algebra <A>KG</A>
rather than that of its associated Lie algebra.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> LieUpperNilpotencyIndex( KG );
5
]]>
</Example>
<ManSection>
<Attr Name="LieLowerNilpotencyIndex"
Arg="KG" />
<Description>
<Index>lower Lie power series</Index>
In a modular group algebra <M>KG</M> the <E>lower
Lie power series</E> is defined as follows:
<M>KG^{[n]}</M> is the associative
ideal, generated by all (left-normed) Lie-products
<M>[x_1, x_2, \dots, x_n]</M>, <M> x_i \in KG </M>.
The lower Lie nilpotency index <M>t_L(G)</M> of the group
algebra <M>KG</M> is defined to be the
minimal smallest <M>n</M> such that <M>KG^{[n]}=0</M>.
In <Cite Key="Du" /> the Jennings' conjecture was proved,
which means that the nilpotency class of the normalized
unit group of the modular group algebra <M>KG</M> is
equal to <M>t_L(G)-1</M>.
<P/>
This allows to express lower Lie nilpotency index via the
nilpotency class of the normalized unit group, and with its
polycyclic presentation, provided by &LAGUNA;, this will be
faster than elementary calculations with Lie commutators.
As the previous attribute, this index is also stored as an
attribute of the group algebra <A>KG</A>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF( 2 ), DihedralGroup( 16 ) );
<algebra-with-one over GF(2), with 4 generators>
gap> LieLowerNilpotencyIndex( KG );
5
]]>
</Example>
<ManSection>
<Attr Name="LieDerivedLength"
Arg="L" />
<Description>
<Index>Lie derived series</Index>
<Index>Lie derived length</Index>
Let <M>L</M> be a Lie algebra. The <E>Lie derived series</E>
of <M>L</M> is defined as follows:
<M>\delta^{[0]}(L) = L</M> and
<M>\delta^{[n]}(L) = [\delta^{[n-1]}(L),
\delta^{[n-1]}(L)]</M>.
<M>L</M> is called Lie solvable if there exists an integer
<M>m</M> such that <M> \delta^{[m]}(L) = 0 </M>. In this
case the integer <M>m</M> is called the <E>Lie derived length</E>
of <M>L</M>, and it is returned by this function.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KG := GroupRing( GF ( 2 ), DihedralGroup( 16 ) );;
gap> L := LieAlgebra( KG );
#I LAGUNA package: Constructing Lie algebra ...
<Lie algebra over GF(2)>
gap> LieDerivedLength( L );
#I LAGUNA package: Computing the Lie derived subalgebra ...
3
]]>
</Example>
</Section>
<!-- ######################################################### -->
<Section Label="Other">
<Heading>Other commands</Heading>
<ManSection>
<Attr Name="SubgroupsOfIndexTwo"
Arg="G"
Comm="for a group" />
<Description>
Returns a list of subgroups of <M>G</M> with index two.
Such subgroups are important for the investigation of
the Lie structure of the group algebra <M>KG</M> in the case
of characteristic 2.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> SubgroupsOfIndexTwo( DihedralGroup( 16) );
[ Group([ f1, f1*f3, f1*f4, f1*f3*f4 ]), Group([ f2, f2*f3, f2*f4, f2*f3*f4 ]),
Group([ f1*f2, f1*f2*f3, f1*f2*f4, f1*f2*f3*f4 ]) ]
]]>
</Example>
<ManSection>
<Meth Name="DihedralDepth"
Arg="U"
Comm="for a finite 2-group" />
<Description>
For a finite 2-group <A>U</A>, the function returns its
<E>dihedral depth</E>, which is defined to be the maximal
number <M>d</M> such that <A>U</A> contains a subgroup
isomorphic to the dihedral
group of order <M>2^{d+1}</M>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> KD8 := GroupRing( GF(2), DihedralGroup( 8 ) );
<algebra-with-one over GF(2), with 3 generators>
gap> UD8 := PcNormalizedUnitGroup( KD8 );
<pc group of size 128 with 7 generators>
gap> DihedralDepth( UD8 );
2
]]>
</Example>
<ManSection>
<Meth Name="DimensionBasis"
Arg="G"
Comm="for a finite p-group" />
<Returns>
record with two components: `dimensionBasis'
(list of group elements) and `weights' (list of weights)
</Returns>
<Description>
For a finite <M>p</M>-group <A>G</A>, returns its
Jennings basis as it was described in Section
<Ref Label="TheoryThird" />.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>
gap> DimensionBasis( G );
rec( dimensionBasis := [ f1, f2, f3, f4 ], weights := [ 1, 1, 2, 4 ] )
]]>
</Example>
<ManSection>
<Attr Name="LieDimensionSubgroups"
Arg="G"
Comm="for a finite p-group" />
<Returns>
list of subgroups
</Returns>
<Description>
For a finite <M>p</M>-group <A>G</A>, returns the series
of its Lie dimension subgroups. The <M>m</M>-th Lie dimension
subgroup <M> D_{(m)} </M> is the intersection of the group
<M>G</M> and <M> 1+KG^{(m)} </M>, where
<M>KG^{(m)}</M> is the <M>m</M>-th term of the
upper Lie power series of <M>KG</M>;
see <Ref Attr="LieUpperNilpotencyIndex" />
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>
gap> LieDimensionSubgroups( G );
[ <pc group of size 16 with 4 generators>, Group([ f3, f4 ]), Group([ f4 ]),
Group([ <identity> of ... ]) ]
]]>
</Example>
<ManSection>
<Attr Name="LieUpperCodimensionSeries"
Arg="KG"
Comm="for a modular group algebra of a finite p-group" />
<Attr Name="LieUpperCodimensionSeries"
Arg="G"
Comm="for a finite p-group" />
<Returns>
list of subgroups
</Returns>
<Description>
A notion of upper Lie codimension subgroups was introduced in
<Cite Key="CS"/>. For a finite <M>p</M>-group <A>G</A>,
<M>C_i</M> is the set of all elements <M>g</M> in <A>G</A>,
such that the Lie commutator <M>[ g, g_1, ..., g_i ]</M> of the
length <M>i+1</M> is equal to zero for all <M>g_1, ..., g_i</M>
from <A>G</A>, and <M> C_0 = {1} </M>.
By Du's theorem (see <Cite Key="Du"/>), <M>C_i</M> coincides
with the intersection of <M>G</M> and the i-th term of the upper
central series
<M>{1}=Z_0 < Z_1 < Z_2 < ... < Z_n = V(KG)</M>
of the normalized unit group <M>V(KG)</M>.
This fact is used in &LAGUNA; to speed up computation of this
series. Since <M>V(KG)</M> is involved in computation, for the
first time the argiment should be the group ring <A>KG</A>, but
later you can also apply it to the group <A>G</A> itself.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> G := DihedralGroup(16);
<pc group of size 16 with 4 generators>
gap> KG := GroupRing( GF(2), G );
<algebra-with-one over GF(2), with 4 generators>
gap> LieUpperCodimensionSeries( KG );
[ Group([ f1, f2, f3, f4 ]), Group([ f3, f4, f3*f4 ]), Group([ f4 ]),
Group([ f4 ]), Group([ ]) ]
gap> LieUpperCodimensionSeries( G );
[ Group([ f1, f2, f3, f4 ]), Group([ f3, f4, f3*f4 ]), Group([ f4 ]),
Group([ f4 ]), Group([ ]) ]
]]>
</Example>
<ManSection>
<InfoClass Name="LAGInfo"
Comm="Info class for LAGUNA algorithms" />
<Description>
<C>LAGInfo</C> is a special Info class for &LAGUNA; algorithms.
It has 5 levels: 0, 1 (default), 2, 3 and 4. To change info
level to <C>k</C>, use command <C>SetInfoLevel(LAGInfo, k)</C>.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> SetInfoLevel( LAGInfo, 2 );
gap> KD8 := GroupRing( GF( 2 ), DihedralGroup( 8 ) );
<algebra-with-one over GF(2), with 3 generators>
gap> UD8 := PcNormalizedUnitGroup( KD8 );
#I LAGInfo: Computing the pc normalized unit group ...
#I LAGInfo: Calculating weighted basis ...
#I LAGInfo: Calculating dimension basis ...
#I LAGInfo: dimension basis finished !
#I LAGInfo: Weighted basis finished !
#I LAGInfo: Computing the augmentation ideal filtration...
#I LAGInfo: Filtration finished !
#I LAGInfo: finished, converting to PcGroup
<pc group of size 128 with 7 generators>
]]>
</Example>
<ManSection>
<Func Name="LAGUNABuildManual"
Arg=""
Comm="requires GAPDoc, UNIX or Linux and TeX" />
<Description>
This function is used to build the manual in the following formats:
DVI, PDF, PS, HTML and text for online help.
We recommend that the user should have a recent and fairly
complete &TeX; distribution.
Since &LAGUNA; is distributed together with its manual,
it is not necessary for the user to use this function. Normally
it is intended to be used by the developers only. This is the only
function of &LAGUNA; which requires UNIX/Linux environment.
</Description>
</ManSection>
<ManSection>
<Func Name="LAGUNABuildManualHTML"
Arg=""
Comm="requires only GAPDoc" />
<Description>
This fuction is used to build the manual only in HTML format.
This does not depend on the availability of the &TeX; installation
and works under Windows and MacOS as well.
Since &LAGUNA; is distributed together with its manual,
it is not necessary for the user to use this function. Normally
it is intended to be used by the developers only.
</Description>
</ManSection>
</Section>
</Chapter>
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91213ddcfbe7f54821d42c2d9e091326
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