|
#############################################################################
##
#W maps.gi GAP4 package `Utils' Stefan Kohl
## Chris Wensley
#Y Copyright (C) 2015-2019, The GAP Group
#############################################################################
##
#F EpimorphismByGenerators( <D1>, <D2> ) . epi: gen's of <D1>->gen's of <D2>
##
InstallMethod( EpimorphismByGenerators, "for groups", ReturnTrue,
[ IsGroup, IsGroup ], 0,
function( G, H )
if not ( IsFreeGroup( G ) ) then
Print( "Warning: calling GroupHomomorphismByImagesNC without checks\n" );
fi;
return GroupHomomorphismByImagesNC( G, H, GeneratorsOfGroup(G),
GeneratorsOfGroup(H) );
end );
##############################################################################
##
#M Pullback . . . . . . . . . for two group homomorphisms with a common range
##
InstallMethod( Pullback, "for two group homomorphisms", true,
[ IsGroupHomomorphism, IsGroupHomomorphism ], 0,
function( nu, mu )
local M, N, P, NxM, projM, pmu, projN, pnu, genNxM, e, genL, L,
imphi, phi, impsi, psi, info;
M := Source( mu );
P := Range( mu );
N := Source( nu );
if not ( Range(nu) = P ) then
Error( "homs nu,mu should have a common range" );
fi;
NxM := DirectProduct( N, M );
genNxM := GeneratorsOfGroup( NxM );
projN := Projection( NxM, 1 );
pnu := projN * nu;
projM := Projection( NxM, 2 );
pmu := projM * mu;
if IsFinite( NxM ) then
genL := [ ];
L := Subgroup( NxM, [ One(NxM) ] );
for e in NxM do
if ImageElm( pnu, e ) = ImageElm( pmu, e ) then
if not ( e in L ) then
Add( genL, e );
L := Group( genL );
fi;
fi;
od;
else
return fail;
fi;
imphi := List( genL, g -> ImageElm( projN, g ) );
phi := GroupHomomorphismByImages( L, N, genL, imphi );
impsi := List( genL, g -> ImageElm( projM, g ) );
psi := GroupHomomorphismByImages( L, M, genL, impsi );
info := rec( directProduct := NxM, projections := [phi,psi] );
SetPullbackInfo( L, info );
return L;
end );
#############################################################################
##
#M CentralProduct( <G1>, <G2>, <Z1>, <Phi> )
##
InstallMethod( CentralProduct,
[ "IsGroup", "IsGroup", "IsGroup", "IsGroupHomomorphism" ],
function( G1, G2, Z1, Phi )
local gens, imgs, dp, emb1, emb2, N, proj, G;
if not ( IsSubset( G1, Z1 ) and IsCentral( G1, Z1 ) ) then
Error( "<Z1> must be a central subgroup of <G1>" );
fi;
gens:= GeneratorsOfGroup( Z1 );
imgs:= List( gens, x -> (x^-1)^Phi );
if not ( IsSubset( G2, imgs ) and
ForAll( imgs, x -> IsCentral( G2, x ) ) ) then
Error( "<Phi> must map <Z1> to a central subgroup of <G2>" );
fi;
dp:= DirectProduct( G1, G2 );
emb1:= Embedding( dp, 1 );
emb2:= Embedding( dp, 2 );
N:= SubgroupNC( dp,
List( [ 1 .. Length( gens ) ], i -> gens[i]^emb1 * imgs[i]^emb2 ) );
proj:= NaturalHomomorphismByNormalSubgroup( dp, N );
G:= Image( proj );
SetCentralProductInfo( G, rec( projection:= proj, phi:= Phi ) );
return G;
end );
##############################################################################
##
#M IdempotentEndomorphisms . . . . . . . . . . . . . . . . . . . for a group
#M IdempotentEndomorphismsData . . . . . . . . . . . . . . . . . for a group
#M IdempotentEndomorphismsWithImage . . . . . for a group and a chosen image
##
InstallMethod( IdempotentEndomorphismsWithImage,
"for a list of group generators and a chosen image",
[ IsList, IsGroup ], 0,
function( genG, R )
local G, numG, r, q, norm, n, reps, i, j, rc;
G := Group( genG );
if not IsSubgroup( G, R ) then
Error( "R should be a subgroup of G" );
fi;
numG := Length( genG );
r := Size( R );
q := Size( G )/r;
norm := Filtered( NormalSubgroups( G ), N -> ( Size( N ) = q ) and
IsTrivial( Intersection( N, R ) ) );
n := Length( norm );
reps := [ ];
for i in [1..n] do
Add( reps, [ ] );
for j in [1..numG] do
rc := norm[i]*genG[j];
Add( reps[i], First( rc, g -> g in R ) );
od;
od;
return reps;
end );
InstallMethod( IdempotentEndomorphismsData, "for a group", [ IsGroup ], 0,
function( G )
local genG, R, data, images;
genG := SmallGeneratingSet( G );
images := [ ];
for R in AllSubgroups( G ) do
data := IdempotentEndomorphismsWithImage( genG, R );
if ( data <> [ ] ) then
Add( images, data );
fi;
od;
return rec( gens := genG, images := images );
end );
InstallMethod( IdempotentEndomorphisms, "for a group", [ IsGroup ], 0,
function( G )
local data, genG, images, len, L, i, im;
data := IdempotentEndomorphismsData( G );
genG := data!.gens;
G := Group( genG );
images := data!.images;
len := Length( images );
L := [ ];
for i in [1..len] do
for im in images[i] do
Add( L, GroupHomomorphismByImages( G, G, genG, im ) );
od;
od;
return L;
end );
##############################################################################
##
#M DirectProductOfFunctions . . . . . . for two groups and two homomorphisms
##
InstallMethod( DirectProductOfFunctions, "for two groups and two homs",
[ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ], 0,
function( G, H, f1, f2 )
local infoG, gpsG, G1, G2, infoH, gpsH, H1, H2, eG1, eG2, eH1, eH2,
mgi1, mgi2, genG, imH1, imH2, imH;
if not HasDirectProductInfo( G ) and HasDirectProductInfo( H ) then
Error( "first two parameters should be direct products" );
fi;
infoG := DirectProductInfo( G );
gpsG := infoG!.groups;
G1 := gpsG[1];
G2 := gpsG[2];
if not ( ( G1 = Source( f1 ) ) and ( G2 = Source( f2 ) ) ) then
Error( "f1,f2 should have source G1,G2" );
fi;
eG1 := Embedding( G, 1 );
eG2 := Embedding( G, 2 );
mgi1 := MappingGeneratorsImages( f1 );
mgi2 := MappingGeneratorsImages( f2 );
infoH := DirectProductInfo( H );
gpsH := infoH!.groups;
H1 := gpsH[1];
H2 := gpsH[2];
if not ( ( H1 = Range( f1 ) ) and ( H2 = Range( f2 ) ) ) then
Error( "f1,f2 should have range H1,H2" );
fi;
eH1 := Embedding( H, 1 );
eH2 := Embedding( H, 2 );
genG := Concatenation( List( mgi1[1], g -> ImageElm( eG1, g ) ),
List( mgi2[1], g -> ImageElm( eG2, g ) ) );
imH1 := List( mgi1[2], h -> ImageElm( eH1, h ) );
imH2 := List( mgi2[2], h -> ImageElm( eH2, h ) );
imH := Concatenation( imH1, imH2 );
return GroupHomomorphismByImagesNC( G, H, genG, imH );
end );
##############################################################################
##
#M DirectProductOfAutomorphismGroups . . . . . . for two automorphism groups
##
InstallMethod( DirectProductOfAutomorphismGroups, "for two groups",
[ IsGroup, IsGroup ], 0,
function( A1, A2 )
local gen1, gen2, G1, G2, id1, id2, dp, dp1, dp2, gen12, A12,
em1, em2, pr1, pr2, info;
if not ( IsGroupOfAutomorphisms(A1) and IsGroupOfAutomorphisms(A2) ) then
Error( "A1,A2 should be automorphism groups" );
fi;
gen1 := GeneratorsOfGroup( A1 );
gen2 := GeneratorsOfGroup( A2 );
G1 := Source( gen1[1] );
G2 := Source( gen2[1] );
if not ( IsGroup( G1 ) and IsGroup( G2 ) ) then
Error( "A1,A2 should be automorphism groups of groups G1,G2" );
fi;
id1 := IdentityMapping( G1 );
id2 := IdentityMapping( G2 );
dp := DirectProduct( G1, G2 );
dp1 := List( gen1, g -> DirectProductOfFunctions( dp, dp, g, id2 ) );
dp2 := List( gen2, g -> DirectProductOfFunctions( dp, dp, id1, g ) );
gen12 := Concatenation( dp1, dp2 );
A12 := Group( gen12 );
em1 := GroupHomomorphismByImages( A1, A12, gen1, dp1 );
em2 := GroupHomomorphismByImages( A2, A12, gen2, dp2 );
pr1 := GroupHomomorphismByImages( A12, A1, gen12,
Concatenation( gen1, List( gen2, g -> id1 ) ) );
pr2 := GroupHomomorphismByImages( A12, A2, gen12,
Concatenation( List( gen1, g -> id2 ), gen2 ) );
info := rec( embeddings := [ em1, em2 ],
groups := [ A1, A2 ],
projections := [ pr1, pr2 ] );
SetDirectProductInfo( A12, info );
return A12;
end );
[ Dauer der Verarbeitung: 0.7 Sekunden
(vorverarbeitet)
]
|
