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##
#W laguna.gd The LAGUNA package Viktor Bovdi
#W Olexandr Konovalov
#W Richard Rossmanith
#W Csaba Schneider
##
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##
## LAGInfo
##
## We declare new Info class for LAGUNA algorithms.
## It has 4 levels - 0, 1 (default), 2 and 3
## To change Info level to k, use command SetInfoLevel(LAGInfo, k)
DeclareInfoClass("LAGInfo");
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##
## SOME CLASSES OF GROUP RINGS AND THEIR GENERAL ATTRIBUTES
##
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##
#P IsGroupAlgebra( <R> )
##
## A group ring over the field is called group algebra. This property
## will be determined automatically for every group ring, created by
## the function `GroupRing'
DeclareProperty("IsGroupAlgebra", IsGroupRing);
InstallTrueMethod(IsGroupRing, IsGroupAlgebra);
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##
#P IsFModularGroupAlgebra( <R> )
##
## A group algebra $FG$ over the field $F$ of characteristic $p$ is called
## modular, if $p$ devides the order of some element of $G$. This property
## will be determined automatically for every group ring, created by
## the function `GroupRing'
DeclareProperty("IsFModularGroupAlgebra", IsGroupAlgebra);
InstallTrueMethod(IsGroupAlgebra, IsFModularGroupAlgebra);
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##
#P IsPModularGroupAlgebra( <R> )
##
## We define separate property for modular group algebras of finite p-groups.
## This property will be determined automatically for every group ring,
## created by the function `GroupRing'
DeclareProperty("IsPModularGroupAlgebra", IsFModularGroupAlgebra);
InstallTrueMethod(IsGroupAlgebra, IsPModularGroupAlgebra);
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##
#A UnderlyingGroup( <R> )
##
## This attribute returns the result of the function `UnderlyingMagma' and
## for convenience was defined for group rings mainly for teaching purposes
DeclareAttribute("UnderlyingGroup", IsGroupRing);
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##
#A UnderlyingRing( <R> )
##
## This attribute returns the result of the function `LeftActingDomain' and
## for convenience was defined for group rings mainly for teaching purposes
DeclareAttribute("UnderlyingRing", IsGroupRing);
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##
#A UnderlyingField( <R> )
##
## This attribute returns the result of the function `LeftActingDomain' and
## for convenience was defined for group algebras mainly for teaching purposes
DeclareAttribute("UnderlyingField", IsGroupAlgebra);
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##
## GENERAL PROPERTIES AND ATTRIBUTES OF GROUP RING ELEMENTS
##
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##
#A Support( <x> )
##
## The support of a non-zero element of a group ring $ x = \sum \alpha_g g $
## is a set of elements $g \in G$ for which $\alpha_g$ in not zero.
## Note that for zero element this function returns an empty list
DeclareAttribute("Support",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#A CoefficientsBySupport( <x> )
##
## List of coefficients for elements of Support(x)
DeclareAttribute("CoefficientsBySupport",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#A TraceOfMagmaRingElement( <x> )
##
## The trace of an element $ x = \sum \alpha_g g $ is $\alpha_1$, i.e.
## the coefficient of the identity element of a group $G$.
## Note that for zero element this function returns zero
DeclareAttribute("TraceOfMagmaRingElement",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#A Length( <x> )
##
## Length of an element of a group ring is the number of elements in its
## support
DeclareAttribute("Length",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#A Augmentation( <x> )
##
## Augmentation of a group ring element $ x = \sum \alpha_g g $ is the sum
## of coefficients $ \sum \alpha_g $
DeclareAttribute("Augmentation",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#O PartialAugmentations( <KG>, <x> )
##
## Returns a list of two lists, the first being partial augmentations of x
## and the second - representatives of corresponding conjugacy classes
DeclareOperation("PartialAugmentations",
[IsGroupRing,
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep ]);
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##
#O Involution( <x> )
##
DeclareOperation("Involution",
[IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep, IsMapping, IsMapping ]);
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##
#A IsSymmetric( <x> )
##
## An element of a group ring is called symmetric if it is fixed under the
## classical involution
DeclareProperty("IsSymmetric",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
#A IsUnitary( <x> )
##
## An unit of a group ring is called unitary if x^-1 = Involution(x) * eps,
## where eps is an invertible element from an underlying ring
DeclareProperty("IsUnitary",
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep );
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##
## AUGMENTATION IDEAL AND GROUPS OF UNITS OF GROUP RINGS
##
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##
#O LeftIdealBySubgroup( <KG>, <H> )
#O RightIdealBySubgroup( <KG>, <H> )
#O TwoSidedIdealBySubgroup( <KG>, <H> )
#O IdealBySubgroup( <KG>, <H> )
##
DeclareOperation( "LeftIdealBySubgroup" , [ IsGroupRing, IsGroup ] );
DeclareOperation( "RightIdealBySubgroup", [ IsGroupRing, IsGroup ] );
DeclareOperation( "TwoSidedIdealBySubgroup", [ IsGroupRing, IsGroup ] );
DeclareSynonym( "IdealBySubgroup", TwoSidedIdealBySubgroup );
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##
#A WeightedBasis( <KG> )
##
## KG must be a modular group algebra. The weighted basis is a basis of the
## fundamental ideal such that each power of the fundamental ideal is
## spanned by a subset of the basis. Note that this function actually
## constructs a basis for the *fundamental ideal* and not for KG.
## Returns a record whose basis entry is the basis and the weights entry
## is a list of corresponding weights of basis elements with respect to
## the fundamental ideal filtration.
## This function uses the Jennings basis of the underlying group.
DeclareAttribute("WeightedBasis", IsPModularGroupAlgebra);
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##
#A AugmentationIdealPowerSeries( <KG> )
##
## KG is a modular group algebra.
## Returns a list whose elements are the terms of the augmentation ideal
## filtration of I. That is AugmentationIdealPowerSeries(KG)[k] = I^k,
## where I is the augmentation ideal of KG.
DeclareAttribute("AugmentationIdealPowerSeries", IsPModularGroupAlgebra);
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##
#A AugmentationIdealNilpotencyIndex( <R> )
##
DeclareAttribute("AugmentationIdealNilpotencyIndex", IsPModularGroupAlgebra);
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##
#A AugmentationIdealOfDerivedSubgroupNilpotencyIndex( <R> )
##
DeclareAttribute("AugmentationIdealOfDerivedSubgroupNilpotencyIndex",
IsPModularGroupAlgebra);
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##
#P IsGroupOfUnitsOfMagmaRing( <U> )
##
DeclareCategory("IsGroupOfUnitsOfMagmaRing", IsGroup);
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##
#P IsUnitGroupOfGroupRing( <U> )
##
DeclareCategory("IsUnitGroupOfGroupRing", IsGroup);
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##
#P IsNormalizedUnitGroupOfGroupRing( <U> )
##
DeclareCategory("IsNormalizedUnitGroupOfGroupRing", IsGroup);
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##
#A UnderlyingGroupRing( <U> )
##
DeclareAttribute("UnderlyingGroupRing", IsGroupOfUnitsOfMagmaRing);
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##
#O NormalizedUnitCF( <KG>, <u> )
##
DeclareOperation( "NormalizedUnitCF",
[ IsPModularGroupAlgebra,
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep] );
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##
#O NormalizedUnitCFmod( <KG>, <u>, <k> )
##
DeclareOperation( "NormalizedUnitCFmod",
[ IsPModularGroupAlgebra,
IsElementOfMagmaRingModuloRelations and
IsMagmaRingObjDefaultRep,
IsPosInt] );
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##
#A NormalizedUnitGroup( <KG> )
##
DeclareAttribute("NormalizedUnitGroup", IsPModularGroupAlgebra);
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#A PcNormalizedUnitGroup( <KG> )
##
DeclareAttribute("PcNormalizedUnitGroup", IsPModularGroupAlgebra);
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##
#O AugmentationIdealPowerFactorGroup( <KG>, <n> )
##
## Calculates the pc-presentation of the factor group of the normalized unit
## group V(KG) over 1+I^n, where I is the augmentation ideal of KG
KeyDependentOperation( "AugmentationIdealPowerFactorGroup",
IsPModularGroupAlgebra, IsPosInt, IsPosInt );
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##
#A PcUnits( <KG> )
##
DeclareAttribute("PcUnits", IsPModularGroupAlgebra);
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##
#A NaturalBijectionToPcNormalizedUnitGroup( <KG> )
##
DeclareAttribute("NaturalBijectionToPcNormalizedUnitGroup",
IsPModularGroupAlgebra);
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##
#A NaturalBijectionToNormalizedUnitGroup( <KG> )
##
DeclareAttribute("NaturalBijectionToNormalizedUnitGroup",
IsPModularGroupAlgebra);
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#A GroupBases( <KG> )
##
DeclareAttribute("GroupBases", IsPModularGroupAlgebra);
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##
#O BassCyclicUnit( <ZG>, <g>, <k> )
#O BassCyclicUnit( <g>, <k> )
##
## Let g be an element of order n of the group G, and 1 < k < n be such that
## k and n are coprime, then k^Phi(n) is congruent to 1 modulo n. The unit
## b(g,k) = ( \sum_{j=0}^{k-1} g^j )^Phi(n) + ( (1-k^Phi(n))/n ) * Hat(g),
## where Hat(g) = g + g^2 + ... + g^n, is called a Bass cyclic unit of
## the integral group ring ZG.
## When G is a finite nilpotent group, the group generated by the
## Bass cyclic units contain a subgroup of finite index in the centre
## of U(ZG) [E. Jespers, M.M. Parmenter and S.K. Sehgal, Central Units
## Of Integral Group Rings Of Nilpotent Groups.
## Proc. Amer. Math. Soc. 124 (1996), no. 4, 1007--1012].
##
DeclareOperation( "BassCyclicUnit",
[ IsGroupRing, IsObject, IsPosInt ] );
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##
#O BicyclicUnitOfType1( <KG>, <a>, <g> )
#O BicyclicUnitOfType2( <KG>, <a>, <g> )
##
## For elements a and g of the underlying group of a group ring KG,
## returns the bicyclic unit u_(a,g) of the appropriate type.
## If ord a = n, then the bicycle unit of the 1st type is defined as
##
## u_{a,g} = 1 + (a-1) * g * ( 1 + a + a^2 + ... +a^{n-1} )
##
## and the bicycle unit of the 2nd type is defined as
##
## v_{a,g} = 1 + ( 1 + a + a^2 + ... +a^{n-1} ) * g * (a-1)
##
## u_{a,g} and v_{a,g} may coincide for some a and g, but in general
## this does not hold.
##
DeclareOperation( "BicyclicUnitOfType1",
[ IsGroupRing, IsObject, IsObject ] );
DeclareOperation( "BicyclicUnitOfType2",
[ IsGroupRing, IsObject, IsObject ] );
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#A BicyclicUnitGroup( <V(KG)> )
##
## KG is a modular group algebra and V(KG) is its normalized unit group.
## Returns the subgroup of V(KG) generated by all bicyclic units u_{g,h}
## and v_{g,h}, where g and h run over the elements of the underlying
## p-group, and h do not belongs to the normalizer of <g> in G.
DeclareAttribute("BicyclicUnitGroup", IsNormalizedUnitGroupOfGroupRing);
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##
#A UnitarySubgroup( <V(KG)> )
##
DeclareAttribute("UnitarySubgroup", IsNormalizedUnitGroupOfGroupRing);
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## LIE PROPERTIES OF GROUP ALGEBRAS
##
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#C IsLieAlgebraByAssociativeAlgebra( <L> )
##
## This category signifies that the Lie algebra is constructed as an
## associated Lie algebra of an associative algebra. (That knowledge
## cannot be obtained later on.)
DeclareCategory( "IsLieAlgebraByAssociativeAlgebra", IsLieAlgebra );
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##
#A UnderlyingAssociativeAlgebra( <L> )
##
## If a Lie algebra is constructed from an associative algebra, it remembers
## this underlying associative algebra as one of its attributes.
DeclareAttribute( "UnderlyingAssociativeAlgebra",
IsLieAlgebraByAssociativeAlgebra );
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##
#P IsLieAlgebraOfGroupRing( <L> )
##
## If a Lie algebra 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 faster algorithms, so this information is stored as a
## property.
DeclareProperty( "IsLieAlgebraOfGroupRing",
IsLieAlgebraByAssociativeAlgebra);
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##
#P IsBasisOfLieAlgebraOfGroupRing( <B> )
##
## A basis has this property if the basis vectors are exactly the images
## of group elements (in sorted oreder).
## A basis can be told that it has this above property. (This is important
## for the speed of the calculation of the structure constants table.)
DeclareProperty( "IsBasisOfLieAlgebraOfGroupRing", IsBasis);
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##
#A UnderlyingGroup( <L> )
##
## The underlying group of a Lie algebra <L> which is constructed from a
## group ring is defined to be the underlying magma of this group ring.
## *Remark:* The underlying field may be accessed by the command
## `LeftActingDomain( <L> )'.
DeclareAttribute( "UnderlyingGroup", IsLieAlgebraOfGroupRing);
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##
#A NaturalBijectionToLieAlgebra( <A> )
##
## The natural linear bijection between the (isomorphic, but not equal,
## see introduction) underlying vector
## spaces of an associative algebra $A$ and its associated Lie algebra is
## stored as an attribute of $A$.
## (Note that this is a vector space isomorphism between two algebras,
## but not an algebra homomorphism in general.)
DeclareAttribute( "NaturalBijectionToLieAlgebra",
IsAlgebra and IsAssociative);
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##
#A NaturalBijectionToAssociativeAlgebra( <L> )
##
## This is the inverse of the linear bijection mentioned above, stored as
## an attribute of the Lie algebra.
DeclareAttribute( "NaturalBijectionToAssociativeAlgebra",
IsLieAlgebraByAssociativeAlgebra);
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##
#O Embedding( <U>, <L> )
##
## Let <U> be a submagma of a group $G$, $A := FG$ be the group ring of $G$
## over some field $F$, and let <L> be the associated Lie algebra of $A$.
## Then `Embedding( <U>, <L> )' returns the obvious mapping $<U> \to <L>$
## (as the composition of the mappings `Embedding( <U>, <A> )' and
## `NaturalBijectionToLieAlgebra( <A> )'~).
## DeclareOperation( "Embedding", [IsGroup, IsLieAlgebraOfGroupRing]);
#############################################################################
##
#A AugmentationHomomorphism( <A> )
##
## Nomen est omen for this attribute of the group ring $<A> := FG$ of a
## group $G$ over some field $F$.
DeclareAttribute( "AugmentationHomomorphism",
IsAlgebraWithOne and IsGroupRing );
#############################################################################
##
#P IsLieMetabelian( <L> )
##
## A Lie algebra is called (Lie) metabelian, if its (Lie) derived subalgebra
## is (Lie) abelian, i.e. its second (Lie) derived subalgebra is trivial.
##
## Note that he prefix `Lie' is consistently used to
## distinguish properties of Lie algebras from the analogous properties
## of groups (or of general algebras). Not using this prefix may
## result in error messages, or even in wrong results without warning.
DeclareProperty( "IsLieMetabelian",
IsLieAlgebra);
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##
#P IsLieCentreByMetabelian( <L> )
##
## A Lie algebra is called (Lie) centre-by-metabelian,
## if its second (Lie) derived
## subalgebra is contained in the (Lie) centre of the Lie algebra.
##
## Note that he prefix `Lie' is consistently used to
## distinguish properties of Lie algebras from the analogous properties
## of groups (or of general algebras). Not using this prefix may
## result in error messages, or even in wrong results without warning.
DeclareProperty( "IsLieCentreByMetabelian",
IsLieAlgebra);
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##
#A LieUpperNilpotencyIndex( <KG> )
##
DeclareAttribute("LieUpperNilpotencyIndex", IsPModularGroupAlgebra);
#############################################################################
##
#A LieLowerNilpotencyIndex( <KG> )
##
DeclareAttribute("LieLowerNilpotencyIndex", IsPModularGroupAlgebra);
#############################################################################
##
#A LieDerivedLength( <KG> )
##
DeclareAttribute("LieDerivedLength", IsLieAlgebra);
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##
## SOME GROUP-THEORETICAL ATTRIBUTES
##
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##
#A SubgroupsOfIndexTwo( <G> )
##
## A list is returned here. (The subgroups of index two in the group <G>
## are important for the Lie structure of the group algebra $F<G>$, in case
## that the underlying field $F$ has characteristic 2.)
DeclareAttribute( "SubgroupsOfIndexTwo", IsGroup);
#############################################################################
##
#A DihedralDepth( <G> )
##
## The dihedral depth of a finite 2-group $G$ is equal to $d$, if the maximal
## size of the dihedral subgroup contained in a group $G$ is $2^(d+1)$
DeclareAttribute( "DihedralDepth", IsGroup);
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##
#A DimensionBasis( <G> )
##
DeclareAttribute("DimensionBasis", IsGroup);
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##
#A LieDimensionSubgroups( <G> )
##
DeclareAttribute("LieDimensionSubgroups", IsGroup);
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##
#A LieUpperCodimensionSeries( <KG> )
#A LieUpperCodimensionSeries( <G> )
##
DeclareAttribute( "LieUpperCodimensionSeries", IsGroupRing);
DeclareAttribute( "LieUpperCodimensionSeries", IsGroup);
#############################################################################
##
#E
##
