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#W lpres.gd The LPRES-package René Hartung
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#C IsElementOfLpGroup
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DeclareCategory( "IsElementOfLpGroup",
IsMultiplicativeElementWithInverse and IsAssociativeElement );
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#C IsElementOfLpGroupCollection
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DeclareCategoryCollections( "IsElementOfLpGroup" );
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#C IsElementOfLpGroupFamily
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DeclareCategoryFamily( "IsElementOfLpGroup" );
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#C IsSubgroupLpGroup
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DeclareCategory( "IsSubgroupLpGroup", IsSubgroupFgGroup );
InstallTrueMethod( IsSubgroupLpGroup, IsGroup and IsElementOfLpGroupCollection);
#InstallTrueMethod( IsSubgroupLpGroup, IsGroup and IsAssocWordCollection );
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#C IsLpGroup
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#DeclareSynonym( "IsLpGroup", IsSubgroupLpGroup and IsWholeFamily );
DeclareSynonym( "IsLpGroup", IsSubgroupLpGroup and IsGroupOfFamily );
#InstallTrueMethod( IsLpGroup, IsGroup and IsElementOfLpGroupCollection );
#InstallTrueMethod( IsLpGroup, IsGroup and IsGroupOfFamily );
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#O ElementOfLpGroup( <fam>, <word> )
##
## If <fam> is the elements-family of an L-presented group and <word> is a
## word in the free generators underlying this L-presented group, this
## operation creates the element with the representative <word> in the
## LpGroup with elements-family <fam>.
##
DeclareOperation( "ElementOfLpGroup",
[ IsElementOfLpGroupFamily, IsAssocWordWithInverse ]);
InstallTrueMethod( IsGeneratorsOfMagmaWithInverses,
IsElementOfLpGroupCollection );
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#P IsInvariantLPresentation ( <G> )
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DeclareProperty( "IsInvariantLPresentation", IsLpGroup );
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#P IsAscendingLPresentation ( <G> )
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DeclareProperty( "IsAscendingLPresentation", IsLpGroup );
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#A FreeGeneratorsOfLpGroup( <G> )
##
## `FreeGeneratorsOfLpGroup' returns the underlying free generators
## corresponding to the generators of the L-presented group <G>.
##
DeclareAttribute( "FreeGeneratorsOfLpGroup", IsLpGroup and IsGroupOfFamily );
DeclareOperation( "FreeGeneratorsOfWholeGroup", [ IsSubgroupLpGroup ] );
DeclareOperation( "FreeGroupOfWholeGroup", [ IsSubgroupLpGroup ] );
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#A FixedRelatorsOfLpGroup( <G> )
##
## returns the fixed relators of the L-presented group <G> as words
## in the free generators obtained from `FreeGeneratorsOfFpGroup'.
##
DeclareAttribute("FixedRelatorsOfLpGroup",IsLpGroup and IsGroupOfFamily);
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#A EndomorphismsOfLpGroup( <G> )
##
## returns the endomorphisms of the L-presented group <G> as group
## homomorphisms of the underlying free group.
##
DeclareAttribute( "EndomorphismsOfLpGroup", IsLpGroup and IsGroupOfFamily );
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#A IteratedRelatorsOfLpGroup( <G> )
##
## returns the iterated relators of the L-presented group <G> as words
## in the free generators obtained from `FreeGeneratorsOfFpGroup'.
##
DeclareAttribute( "IteratedRelatorsOfLpGroup", IsLpGroup and IsGroupOfFamily );
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#A FreeGroupOfLpGroup(<G>)
##
## returns the underlying free group of the L-presented group <G>; i.e.
## the group generated by the free generators obtained from
## `FreeGeneratorsOfFpGroup'.
##
DeclareAttribute( "FreeGroupOfLpGroup", IsLpGroup and IsGroupOfFamily );
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#F LPresentedGroup ( <FreeGroup>, <rels>, <endos>, <itrels> )
##
## constructs the L-presented group on the generators of <FreeGroup> with
## the fixed relators <rels>, the free group endomorphisms <endos>,
## and the iterated relators <itrels>.
##
DeclareGlobalFunction( "LPresentedGroup" );
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#A UnderlyingAscendingLPresentation( <LpGroup> )
##
## returns the underlying ascending L-presentation.
##
DeclareAttribute( "UnderlyingAscendingLPresentation", IsLpGroup );
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#A UnderlyingInvariantLPresentation( <LpGroup> )
##
## attempts to compute an underlying invariant L-presentation for the
## group <LpGroup>. In the worst case, this routine returns the underlying
## ascending L-presentation. This attribute can be set manually by the user
## using `SetUnderlyingInvariantLPresentation'.
##
DeclareAttribute( "UnderlyingInvariantLPresentation", IsLpGroup );
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#A EmbeddingOfAscendingSubgroup ( <LpGroup> )
##
## an attribute which stores an embedding of an ascending L-presented
## subgroup of the L-presented group <LpGroup>. This is useful for the
## construction in FR of the Gupta-Sidki-Groups with its invariantly
## L-presented index-3 subgroup.
##
DeclareAttribute( "EmbeddingOfAscendingSubgroup", IsLpGroup );
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#O EpimorphismFromFpGroup ( <LpGroup>, <n> )
##
## returns an epimorphism from a finitely presented group G which is obtained
## from <LpGroup> by applying word of length at most <n> in the endomorphisms
## of the L-presented group <LpGroup>
##
DeclareOperation( "EpimorphismFromFpGroup", [ IsLpGroup, IsPosInt ] );
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#O SplitExtensionByAutomorphismsLpGroup ( <G>, <H>, <auts> )
##
## returns the split extension of <G> by <H> where the action of each
## generator of <H> on <G> is given by the automorphisms <aut>: <H> -> <G>
## in the list <auts>.
##
DeclareOperation( "SplitExtensionByAutomorphismsLpGroup",
[ IsLpGroup, IsGroup, IsList ]);
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#O AsLpGroup ( <G> )
##
## returns an L-presented group which is isomorphic to <G>
##
DeclareOperation( "AsLpGroup", [ IsGroup ]);
#DeclareOperation( "AsLpGroup", [ IsFreeGroup ]);
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#O IsomorphismLpGroup ( <G> )
##
## returns an isomorphism from an FpGroup (or from a FreeGroup) to the LpGroup
## obtained from `AsLpGroup'.
##
DeclareAttribute( "IsomorphismLpGroup", IsGroup );
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#P IsFinitelyPresentable( <LpGroup> )
##
## stores if <LpGroup> is finitely presentable; e.g. if <LpGroup> is returned
## by IsomorphismLpGroup applied to an FpGroup.
##
DeclareProperty( "IsFinitelyPresentable", IsGroup );
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#A CoveringGroups( <LpGroup> ) . . . . . stores the (nilp.) covering groups
##
DeclareAttribute( "CoveringGroups", IsLpGroup and
HasIsInvariantLPresentation and IsInvariantLPresentation );
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#O EpimorphismCoveringGroups( <LpGroup>, <int>, <int> )
##
DeclareOperation( "EpimorphismCoveringGroups", [ IsLpGroup and
HasIsInvariantLPresentation and IsInvariantLPresentation,
IsPosInt, IsPosInt ] );
