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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the methods for the construction of the basic fp group
## types.
##
#############################################################################
##
#M TrivialGroupCons( <IsPcGroup> )
##
InstallMethod( TrivialGroupCons, "fp group",
[ IsFpGroup and IsTrivial ],
filter -> FreeGroup(0));
#############################################################################
##
#M AbelianGroupCons( <IsFpGroup and IsFinite>, <ints> )
##
InstallMethod( AbelianGroupCons, "fp group", true,
[ IsFpGroup and IsAbelian, IsList ], 0,
function( filter, ints )
local f,g,i,j,rels,gfam,fam;
if not ForAll( ints, x -> IsInfinity(x) or (IsInt(x) and x >= 0) ) then
Error( "<ints> must be a list of integers" );
fi;
f := FreeGroup(IsSyllableWordsFamily, Length(ints));
g := GeneratorsOfGroup(f);
rels:=[];
for i in [1..Length(ints)] do
for j in [1..i-1] do
Add(rels,Comm(g[i],g[j]));
od;
if IsPosInt(ints[i]) then
Add(rels,g[i]^ints[i]);
fi;
od;
g:=f/rels;
if ForAll(ints,IsPosInt) then
SetSize( g, Product(ints) );
else
SetSize( g, infinity );
fi;
fam:=FamilyObj(One(f));
gfam:=FamilyObj(One(g));
gfam!.redorders:=ints;
SetFpElementNFFunction(gfam,function(x)
local u,e,i,j,n;
u:=UnderlyingElement(x);
e:=ExtRepOfObj(u); # syllable form
# bring in correct order and reduction
n:=ListWithIdenticalEntries(Length(gfam!.redorders),0);
for i in [1,3..Length(e)-1] do
j:=e[i];
if IsPosInt(gfam!.redorders[j]) then
n[j]:=n[j]+e[i+1] mod gfam!.redorders[j];
else
n[j]:=n[j]+e[i+1];
fi;
od;
e:=[];
for i in [1..Length(gfam!.redorders)] do
if n[i]<>0 then
Add(e,i);
Add(e,n[i]);
fi;
od;
return ObjByExtRep(fam,e);
end);
SetReducedMultiplication(g);
SetIsAbelian( g, true );
return g;
end );
#############################################################################
##
#M CyclicGroupCons( <IsFpGroup>, <n> )
##
InstallOtherMethod( CyclicGroupCons, "fp group", true,
[ IsFpGroup and IsCyclic, IsObject ], 0,
function( filter, n )
local f,g,fam,gfam;
if n=infinity then
return FreeGroup("a");
elif not IsPosInt(n) then
TryNextMethod();
fi;
f:=FreeGroup( IsSyllableWordsFamily, "a" );
g:=f/[f.1^n];
SetSize(g,n);
fam:=FamilyObj(One(f));
gfam:=FamilyObj(One(g));
SetFpElementNFFunction(gfam,function(x)
local u,e;
u:=UnderlyingElement(x);
e:=ExtRepOfObj(u); # syllable form
if Length(e)=0 or (e[2]>=0 and e[2]<n) then
return u;
elif e[2] mod n=0 then
return One(f);
else
e:=[e[1],e[2] mod n];
return ObjByExtRep(fam,e);
fi;
end);
SetReducedMultiplication(g);
return g;
end );
#############################################################################
##
#M DihedralGroupCons( <IsFpGroup and IsFinite>, <n> )
##
InstallMethod( DihedralGroupCons,
"fp group",
true,
[ IsFpGroup and IsFinite,
IsInt and IsPosRat ],
0,
function( filter, n )
local f,rels,g;
if n mod 2 = 1 then
TryNextMethod();
elif n = 2 then return
CyclicGroup( IsFpGroup, 2 );
fi;
f := FreeGroup( IsSyllableWordsFamily, "r", "s" );
rels:= [f.1^(n/2),f.2^2,f.1^f.2*f.1];
g := f/rels;
SetSize(g,n);
SetReducedMultiplication(g);
return g;
end );
InstallOtherMethod( DihedralGroupCons,
"fp group",
true,
[ IsFpGroup and IsFinite,
IsInfinity ],
0,
function( filter, inf )
local f,rels,g;
f := FreeGroup( IsSyllableWordsFamily, "r", "s" );
rels:= [f.2^2,f.1^f.2*f.1];
g := f/rels;
SetSize(g,infinity);
SetReducedMultiplication(g);
return g;
end );
#############################################################################
##
#M DicyclicGroupCons( <IsFpGroup and IsFinite>, <n> )
##
InstallMethod( DicyclicGroupCons,
"fp group",
true,
[ IsFpGroup and IsFinite,
IsInt and IsPosRat ],
0,
function( filter, n )
local f,rels,g;
if 0 <> n mod 4 then
TryNextMethod();
elif n = 4 then return
CyclicGroup( IsFpGroup, 4 );
fi;
f := FreeGroup( IsSyllableWordsFamily, "r", "s" );
rels:= [ f.1^2/f.2^(n/4), f.2^(n/2), f.2^f.1*f.2 ];
g := f/rels;
SetSize(g,n);
if n <= 10^4 then SetReducedMultiplication(g); fi;
return g;
end );
#############################################################################
##
#M ElementaryAbelianGroupCons( <IsFpGroup and IsFinite>, <n> )
##
InstallMethod( ElementaryAbelianGroupCons,
"fp group",
true,
[ IsFpGroup and IsFinite and IsElementaryAbelian,
IsInt and IsPosRat ],
0,
function( filter, n )
if n = 1 then
return CyclicGroupCons( IsFpGroup, 1 );
elif not IsPrimePowerInt(n) then
Error( "<n> must be a prime power" );
fi;
n:= AbelianGroupCons( IsFpGroup, Factors(n) );
SetIsElementaryAbelian( n, true );
return n;
end );
#############################################################################
##
#M FreeAbelianGroupCons( <IsFpGroup>, <rank> )
##
InstallMethod( FreeAbelianGroupCons,
"fp group",
true,
[ IsFpGroup and IsAbelian,
IsInt and IsPosRat ],
0,
function( filter, rank )
return AbelianGroupCons( filter, ListWithIdenticalEntries(rank, 0) );
# TODO: Add the following if it ever moves from Polycyclic to the GAP core:
#SetIsFreeAbelian( G, true );
end );
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