Quelle stdgen.gd Sprache: unbekannt

#############################################################################
##
#W stdgen.gd GAP library Thomas Breuer
##
#Y (C) 1999 School Math and Comp. Sci., University of St. Andrews, Scotland
##
## This file contains the declarations needed for dealing with standard
## generators of finite groups.
##

#T TO DO:
#T - a function that can be used to *define* standard generators,
#T using the character table with underlying group (perhaps also the
#T table of marks or an explicit description of all maximal subgroups)

#############################################################################
##
## Standard Generators of Groups
## <#GAPDoc Label="[1]{stdgen}">
## An <M>s</M>-tuple of <E>standard generators</E> of a given group <M>G</M>
## is a vector <M>(g_1, g_2, \ldots, g_s)</M> of elements <M>g_i \in G</M>
## satisfying certain conditions (depending on the isomorphism type of
## <M>G</M>) such that
## <Enum>
## <Item>
## <M>\langle g_1, g_2, \ldots, g_s \rangle = G</M> and
## </Item>
## <Item>
## the vector is unique up to automorphisms of <M>G</M>,
## i.e., for two vectors <M>(g_1, g_2, \ldots, g_s)</M> and
## <M>(h_1, h_2, \ldots, h_s)</M> of standard generators,
## the map <M>g_i \mapsto h_i</M> extends to an automorphism of <M>G</M>.
## </Item>
## </Enum>
## For details about standard generators, see <Cite Key="Wil96"/>.
## <#/GAPDoc>
##

#############################################################################
##
#A StandardGeneratorsInfo( <G> )
##
## <#GAPDoc Label="StandardGeneratorsInfo:stdgen">
## <ManSection>
## <Attr Name="StandardGeneratorsInfo" Arg='G' Label="for groups"/>
##
## <Description>
## When called with the group <A>G</A>,
## <Ref Func="StandardGeneratorsInfo" Label="for groups"/> returns a list of
## records with at least one of the components <C>script</C> and
## <C>description</C>.
## Each such record defines <E>standard generators</E> of groups isomorphic
## to <A>G</A>, the <M>i</M>-th record is referred to as the <M>i</M>-th set
## of standard generators for such groups.
## The value of <C>script</C> is a dense list of lists, each encoding a
## command that has one of the following forms.
## <List>
## A <E>definition</E> <M>[ i, n, k ]</M> or <M>[ i, n ]</M>
## <Item>
## means to search for an element of order <M>n</M>,
## and to take its <M>k</M>-th power as candidate for the <M>i</M>-th
## standard generator (the default for <M>k</M> is <M>1</M>),
## </Item>
## a <E>relation</E> <M>[ i_1, k_1, i_2, k_2, \ldots, i_m, k_m, n ]</M> with <M>m > 1</M>
## <Item>
## means a check whether the element
## <M>g_{{i_1}}^{{k_1}} g_{{i_2}}^{{k_2}} \cdots g_{{i_m}}^{{k_m}}</M>
## has order <M>n</M>; if <M>g_j</M> occurs then of course the
## <M>j</M>-th generator must have been defined before,
## </Item>
## a <E>relation</E> <M>[ [ i_1, i_2, \ldots, i_m ], <A>slp</A>, n ]</M>
## <Item>
## means a check whether the result of the straight line program
## <A>slp</A> (see <Ref Sect="Straight Line Programs" BookName="ref"/>) applied to
## the candidates <M>g_{{i_1}}, g_{{i_2}}, \ldots, g_{{i_m}}</M> has
## order <M>n</M>, where the candidates <M>g_j</M> for the <M>j</M>-th
## standard generators must have been defined before,
## </Item>
## a <E>condition</E> <M>[ [ i_1, k_1, i_2, k_2, \ldots, i_m, k_m ], f, v ]</M>
## <Item>
## means a check whether the &GAP; function in the global list
## <Ref Var="StandardGeneratorsFunctions"/>
## that is followed by the list <M>f</M> of strings returns the value
## <M>v</M> when it is called with <M>G</M> and
## <M>g_{{i_1}}^{{k_1}} g_{{i_2}}^{{k_2}} \cdots g_{{i_m}}^{{k_m}}</M>.
## </Item>
## </List>
## Optional components of the returned records are
## <List>
## <C>generators</C>
## <Item>
## a string of names of the standard generators,
## </Item>
## <C>description</C>
## <Item>
## a string describing the <C>script</C> information in human readable
## form, in terms of the <C>generators</C> value,
## </Item>
## <C>classnames</C>
## <Item>
## a list of strings, the <M>i</M>-th entry being the name of the
## conjugacy class containing the <M>i</M>-th standard generator,
## according to the &ATLAS; character table of the group
## (see <Ref Func="ClassNames" BookName="ref"/>), and
## 
## 
## </Item>
## <C>ATLAS</C>
## <Item>
## a boolean; <K>true</K> means that the standard generators coincide
## with those defined in Rob Wilson's &ATLAS; of Group Representations
## (see <Cite Key="AGR"/>), and <K>false</K> means that this
## property is not guaranteed.
## </Item>
## <C>standardization</C>
## <Item>
## a positive integer <M>i</M>; Whenever <C>ATLAS</C> returns <K>true</K> the value of <M>i</M> means that the generators
## stored in the group are standard generators w.r.t. standardization <M>i</M>, in the sense of Rob Wilson's &ATLAS; of Group Representations.
## </Item>
## </List>
## 
## There is no default method for an arbitrary isomorphism type,
## since in general the definition of standard generators is not obvious.
## 
## The function <Ref Func="StandardGeneratorsOfGroup"/>
## can be used to find standard generators of a given group isomorphic
## to <A>G</A>.
## 
## The <C>generators</C> and <C>description</C> values, if not known,
## can be computed by <Ref Func="HumanReadableDefinition"/>.
## <Example><![CDATA[
## gap> StandardGeneratorsInfo( TableOfMarks( "L3(3)" ) );
##[ rec( ATLAS := true,
## description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4",
## generators := "a, b",
## script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
## [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ],
## standardization := 1 ) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "StandardGeneratorsInfo", IsGroup );
#T make this an operation also for strings?

#############################################################################
##
#F HumanReadableDefinition( <info> )
#F ScriptFromString( <string> )
##
## <#GAPDoc Label="HumanReadableDefinition">
## <ManSection>
## <Func Name="HumanReadableDefinition" Arg='info'/>
## <Func Name="ScriptFromString" Arg='string'/>
##
## <Description>
## Let <A>info</A> be a record that is valid as value of
## <Ref Func="StandardGeneratorsInfo" Label="for groups"/>.
## <Ref Func="HumanReadableDefinition"/> returns a string that describes the
## definition of standard generators given by the <C>script</C> component of
## <A>info</A> in human readable form.
## The names of the generators are taken from the <C>generators</C>
## component (default names <C>"a"</C>, <C>"b"</C> etc. are computed
## if necessary),
## and the result is stored in the <C>description</C> component.
## 
## <Ref Func="ScriptFromString"/> does the converse of
## <Ref Func="HumanReadableDefinition"/>, i.e.,
## it takes a string <A>string</A> as returned by
## <Ref Func="HumanReadableDefinition"/>, and returns a corresponding
## <C>script</C> list.
## 
## If <Q>condition</Q> lines occur in the script
## (see <Ref Func="StandardGeneratorsInfo" Label="for groups"/>)
## then the functions that occur must be contained in
## <Ref Var="StandardGeneratorsFunctions"/>.
## <Example><![CDATA[
## gap> scr:= ScriptFromString( "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4" );
## [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ], [ 1, 1, 2, 1, 13 ],
## [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ]
## gap> info:= rec( script:= scr );
## rec( script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
## [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
## gap> HumanReadableDefinition( info );
## "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4"
## gap> info;
## rec( description := "|a|=2, |b|=3, |C(b)|=9, |ab|=13, |ababb|=4",
## generators := "a, b",
## script := [ [ 1, 2 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 9 ],
## [ 1, 1, 2, 1, 13 ], [ 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 ] ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "HumanReadableDefinition" );

DeclareGlobalFunction( "ScriptFromString" );

#############################################################################
##
#V StandardGeneratorsFunctions
##
## <#GAPDoc Label="StandardGeneratorsFunctions">
## <ManSection>
## <Var Name="StandardGeneratorsFunctions"/>
##
## <Description>
## <Ref Func="StandardGeneratorsFunctions"/> is a list of even length.
## At position <M>2i-1</M>, a function of two arguments is stored,
## which are expected to be a group and a group element.
## At position <M>2i</M> a list of strings is stored such that first
## inserting a generator name in all holes and then forming the
## concatenation yields a string that describes the function at the previous
## position;
## this string must contain the generator enclosed in round brackets
## <C>(</C> and <C>)</C>.
## 
## This list is used by the functions
## <Ref Func="StandardGeneratorsInfo" Label="for groups"/>),
## <Ref Func="HumanReadableDefinition"/>, and
## <Ref Func="ScriptFromString"/>.
## Note that the lists at even positions must be pairwise different.
## <Example><![CDATA[
## gap> StandardGeneratorsFunctions{ [ 1, 2 ] };
## [ function( G, g ) ... end, [ "|C(",, ")|" ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
BindGlobal( "StandardGeneratorsFunctions", [] );

#############################################################################
##
#F IsStandardGeneratorsOfGroup( <info>, <G>, <gens> )
##
## <#GAPDoc Label="IsStandardGeneratorsOfGroup">
## <ManSection>
## <Func Name="IsStandardGeneratorsOfGroup" Arg='info, G, gens'/>
##
## <Description>
## Let <A>info</A> be a record that is valid as value of
## <Ref Func="StandardGeneratorsInfo" Label="for groups"/>,
## <A>G</A> a group, and <A>gens</A> a list of generators for <A>G</A>.
## In this case, <Ref Func="IsStandardGeneratorsOfGroup"/> returns
## <K>true</K> if <A>gens</A> satisfies the conditions of the <C>script</C>
## component of <A>info</A>, and <K>false</K> otherwise.
## 
## Note that the result <K>true</K> means that <A>gens</A> is a list of
## standard generators for <A>G</A> only if <A>G</A> has the isomorphism
## type for which <A>info</A> describes standard generators.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsStandardGeneratorsOfGroup" );

#############################################################################
##
#F StandardGeneratorsOfGroup( <info>, <G>[, <randfunc>] )
##
## <#GAPDoc Label="StandardGeneratorsOfGroup">
## <ManSection>
## <Func Name="StandardGeneratorsOfGroup" Arg='info, G[, randfunc]'/>
##
## <Description>
## Let <A>info</A> be a record that is valid as value of
## <Ref Func="StandardGeneratorsInfo" Label="for groups"/>,
## and <A>G</A> a group of the isomorphism type for which <A>info</A>
## describes standard generators.
## In this case, <Ref Func="StandardGeneratorsOfGroup"/> returns a list of
## standard generators of <A>G</A>,
## see Section <Ref Sect="Standard Generators of Groups"/>.
## 
## The optional argument <A>randfunc</A> must be a function that returns an
## element of <A>G</A> when called with <A>G</A>; the default is
## <Ref Func="PseudoRandom" BookName="ref"/>.
## 
## In each call to <Ref Func="StandardGeneratorsOfGroup"/>,
## the <C>script</C> component of <A>info</A> is scanned line by line.
## <A>randfunc</A> is used to find an element of the prescribed order
## whenever a definition line is met,
## and for the relation and condition lines in the <C>script</C> list,
## the current generator candidates are checked;
## if a condition is not fulfilled, all candidates are thrown away,
## and the procedure starts again with the first line.
## When the conditions are fulfilled after processing the last line
## of the <C>script</C> list, the standard generators are returned.
## 
## 
## 
## 
## 
## 
## Note that if <A>G</A> has the wrong isomorphism type then
## <Ref Func="StandardGeneratorsOfGroup"/> returns a list of elements in
## <A>G</A> that satisfy the conditions of the <C>script</C> component of
## <A>info</A> if such elements exist, and does not terminate otherwise.
## In the former case, obviously the returned elements need not be standard
## generators of <A>G</A>.
## <Example><![CDATA[
## gap> a5:= AlternatingGroup( 5 );
## Alt( [ 1 .. 5 ] )
## gap> info:= StandardGeneratorsInfo( TableOfMarks( "A5" ) )[1];
## rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5",
## generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ],
## standardization := 1 )
## gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (3,4,5) ] );
## true
## gap> IsStandardGeneratorsOfGroup( info, a5, [ (1,3)(2,4), (1,2,3) ] );
## false
## gap> s5:= SymmetricGroup( 5 );;
## gap> RepresentativeAction( s5, [ (1,3)(2,4), (3,4,5) ],
## > StandardGeneratorsOfGroup( info, a5 ), OnPairs ) <> fail;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "StandardGeneratorsOfGroup" );

#############################################################################
##
#A StandardGeneratorsInfo( <tom> )
##
## <#GAPDoc Label="StandardGeneratorsInfo:tom">
## <ManSection>
## <Attr Name="StandardGeneratorsInfo" Arg='tom'
## Label="for tables of marks"/>
##
## <Description>
## For a table of marks <A>tom</A>, a stored value of
## <Ref Func="StandardGeneratorsInfo" Label="for tables of marks"/>
## equals the value of this attribute for the underlying group
## (see <Ref Attr="UnderlyingGroup" Label="for tables of marks" BookName="ref"/>)
## of <A>tom</A>,
## cf. Section <Ref Sect="Standard Generators of Groups"/>.
## 
## In this case, the <Ref Func="GeneratorsOfGroup" BookName="ref"/> value of the underlying
## group <M>G</M> of <A>tom</A> is assumed to be in fact a list of
## standard generators for <M>G</M>;
## So one should be careful when setting the
## <Ref Func="StandardGeneratorsInfo" Label="for tables of marks"/> value
## by hand.
## 
## There is no default method to compute the
## <Ref Func="StandardGeneratorsInfo" Label="for tables of marks"/> value
## of a table of marks if it is not yet stored.
## 
## <Example><![CDATA[
## gap> a5:=TableOfMarks("A5");
## gap> std:= StandardGeneratorsInfo( a5 );
## [ rec( ATLAS := true, description := "|a|=2, |b|=3, |ab|=5",
## generators := "a, b", script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ]
## ], standardization := 1 ) ]
## gap> # Now find standard generators of an isomorphic group.
## gap> g:= SL(2,4);;
## gap> repeat
## > x:= PseudoRandom( g );
## > until Order( x ) = 2;
## gap> repeat
## > y:= PseudoRandom( g );
## > until Order( y ) = 3 and Order( x*y ) = 5;
## gap> # Compute a representative w.r.t. these generators.
## gap> RepresentativeTomByGenerators( a5, 4, [ x, y ] );
## Group([ [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ],
## [ [ Z(2^2), Z(2^2)^2 ], [ Z(2^2)^2, Z(2^2) ] ] ])
## gap> # Show that the new generators are really good.
## gap> grp:= UnderlyingGroup( a5 );;
## gap> iso:= GroupGeneralMappingByImages( grp, g,
## > GeneratorsOfGroup( grp ), [ x, y ] );;
## gap> IsGroupHomomorphism( iso );
## true
## gap> IsBijective( iso );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "StandardGeneratorsInfo", IsTableOfMarks );

#############################################################################
##
#E

Quelle stdgen.gd Sprache: unbekannt

[ Dauer der Verarbeitung: 0.33 Sekunden (vorverarbeitet) ]