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############################################################################
##
#W apps.gi GAP4 package `XMod' Chris Wensley
##
#Y Copyright (C) 2001-2024, Chris Wensley et al,
############################################################################
##
InstallMethod( LoopClass, "for a crossed module and an element", true,
[ IsPreXMod, IsObject ], 0,
function( X0, a )
local M0, P0, bdy0, Q0, class, p, c, q, b, len;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
P0 := Range( X0 );
if not ( a in P0 ) then
Error( "element a not in the range group P0" );
fi;
bdy0 := Boundary( X0 );
Q0 := Image( bdy0 );
class := [ ];
for p in P0 do
c := a^p;
for q in Q0 do
b := c*q;
if not ( b in class ) then
Add( class, b );
fi;
od;
od;
len := Length( class );
Print( "class has length ", len, " with elements\n", class, "\n" );
return class;
end );
############################################################################
##
InstallMethod( LoopClasses, "for a crossed module", true, [ IsPreXMod ], 0,
function( X0 )
local M0, P0, bdy0, Q0, classes, nc, class, a, found, i, p, c, q, b, len;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
P0 := Range( X0 );
bdy0 := Boundary( X0 );
Q0 := Image( bdy0 );
classes := [ ];
nc := 0;
for a in P0 do
found := false;
for i in [1..nc] do
if ( a in classes[i] ) then
found := true;
fi;
od;
if not found then
class := [ ];
for p in P0 do
c := a^p;
for q in Q0 do
b := c*q;
if not ( b in class ) then
Add( class, b );
fi;
od;
od;
len := Length( class );
Info( InfoXMod, 1, "class has length ", len,
" with elements\n", class );
Add( classes, class );
nc := nc+1;
fi;
od;
return classes;
end );
InstallMethod( LoopClassOld, "for a crossed module and an element", true,
[ IsPreXMod, IsObject ], 0,
function( X0, a )
local M0, genM0, lenM0, P0, bdy0, act0, aut0, acta, elPa, p, c, m,
nPa, C0, G0, e1, e2, Pa, genPa, posPa, i, e, e2m, e1p, g, ngPa,
imda, j, pa, ma, bdy1, imact1, act1;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
genM0 := GeneratorsOfGroup( M0 );
lenM0 := Length( genM0 );
P0 := Range( X0 );
if not ( a in P0 ) then
Error( "element a not in the range group P0" );
fi;
bdy0 := Boundary( X0 );
act0 := XModAction( X0 );
aut0 := Range( act0 );
acta := ImageElm( act0, a );
elPa := [ ];
for p in P0 do
c := Comm( p, a );
for m in M0 do
if ( ImageElm( bdy0, m ) = c ) then
Add( elPa, [p,m] );
fi;
od;
od;
nPa := Length( elPa );
Print( "elPa has length ", nPa, "\n" );
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( " with elements ", elPa, "\n" );
fi;
C0 := PreCat1GroupRecordOfPreXMod( X0 ).precat1;
G0 := Source( C0 );
e1 := Embedding( G0, 1 );
e2 := Embedding( G0, 2 );
Pa := Subgroup( G0, [ ] );
genPa := [ ];
posPa := [ ];
for i in [1..nPa] do
e := elPa[i];
p := e[1];
m := e[2];
e2m := ImageElm( e2, m );
e1p := ImageElm( e1, p );
g := e1p * e2m;
if not ( g in Pa ) then
Add( genPa, g );
Add( posPa, i );
Pa := Subgroup( G0, genPa );
fi;
od;
Info( InfoXMod, 2, "Pa has size ", Size( Pa ) );
ngPa := Length( genPa );
## construct the boundary for X1
imda := ListWithIdenticalEntries( lenM0, 0 );
for j in [1..lenM0] do
m := genM0[j];
pa := ImageElm( bdy0, m );
ma := m^-1 * ImageElm( acta, m );
e2m := ImageElm( e2, ma );
e1p := ImageElm( e1, pa );
g := e1p * e2m;
if not ( g in Pa ) then
Error( "element g not in Pa" );
fi;
imda[j] := g;
od;
bdy1 := GroupHomomorphismByImages( M0, Pa, genM0, imda );
if ( bdy1 = fail ) then
Error( "bdy1 fails to be a homomorphism" );
fi;
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( "the boundary map:\n" );
Display( bdy1 );
fi;
## construct the action map for X1
imact1 := ListWithIdenticalEntries( ngPa, 0 );
for i in [1..ngPa] do
p := elPa[ posPa[i] ][1];
imact1[i] := ImageElm( act0, p );
od;
act1 := GroupHomomorphismByImages( Pa, aut0, genPa, imact1 );
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( "the action map:\n" );
Display( act1 );
fi;
## now we can construct the crossed module
return XModByBoundaryAndAction( bdy1, act1 );
end );
############################################################################
##
InstallMethod( LoopClassesNew, "for a crossed module", true,
[ IsPreXMod ], 0,
function( X0 )
local M0, iterM0, P0, iterPa, iterPb, iterPp, lenP, elP0,
act0, bdy0, Q0, found, L, i, a, p, m, b,
classes, class, conj, numj, pos, c, c1, j, cj;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
P0 := Range( X0 );
bdy0 := Boundary( X0 );
Q0 := Image( bdy0 );
if IsAbelian( P0 ) then
return List( RightCosets( P0, Q0 ), Representative );
fi;
iterM0 := Iterator( M0 );
## elM0 := Elements( M0 );
iterPa := Iterator( P0 );
iterPp := Iterator( P0 );
elP0 := Elements( P0 );
lenP := Size( P0 );
act0 := XModAction( X0 );
## elQ0 := Elements( Q0 );
found := ListWithIdenticalEntries( lenP, 0 );
L := [ ];
classes := [ ];
while not IsDoneIterator( iterPa ) do
a := NextIterator( iterPa );
if ( found[i] = 0 ) then
found[i] := 1;
class := [ ];
while not IsDoneIterator( iterPp ) do
p := NextIterator( iterPp );
while not IsDoneIterator( iterM0 ) do
m := NextIterator( iterM0 );
b := a^p * ImageElm( bdy0, m ); ## a' = a^p + delta(m)
if not ( b in class ) then
found[ Position( elP0, b ) ] := 1;
Add( class, b );
fi;
od;
od;
Add( classes, class );
fi;
od;
if not IsAbelian( P0 ) then
conj := List( ConjugacyClasses( P0 ), Elements );
numj := Length( conj );
pos := ListWithIdenticalEntries( numj, 0 );
for i in [1..numj] do
c := conj[i];
c1 := c[1];
cj := First( classes, j -> c1 in j );
if ForAll( c, g -> g in cj ) then
pos[i] := Position( classes, cj );
fi;
od;
Print( "positions of conjugacy classes in loop classes: ", pos, "\n" );
fi;
return classes;
end );
############################################################################
##
InstallMethod( LoopClassRepresentatives, "for a crossed module", true,
[ IsPreXMod ], 0,
function( X0 )
local M0, iterM0, P0, iterPa, iterPb, iterPp, lenP,
act0, bdy0, Q0, found, L, i, a, p, m, b,
reps, conj, numj, pos, c, c1, j, cj;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
P0 := Range( X0 );
bdy0 := Boundary( X0 );
Q0 := Image( bdy0 );
if IsAbelian( P0 ) then
return List( RightCosets( P0, Q0 ), Representative );
fi;
iterM0 := Iterator( M0 );
## elM0 := Elements( M0 );
iterPa := Iterator( P0 );
iterPp := Iterator( P0 );
## elP0 := Elements( P0 );
lenP := Size( P0 );
act0 := XModAction( X0 );
## elQ0 := Elements( Q0 );
found := ListWithIdenticalEntries( lenP, 0 );
L := [ ];
reps := [ ];
while not IsDoneIterator( iterPa ) do
a := NextIterator( iterPa );
if ( found[i] = 0 ) then
found[i] := 1;
Add( reps, a );
while not IsDoneIterator( iterPp ) do
p := NextIterator( iterPp );
while not IsDoneIterator( iterM0 ) do
m := NextIterator( iterM0 );
b := a^p * ImageElm( bdy0, m ); ## a' = a^p + delta(m)
od;
od;
fi;
od;
return reps;
end );
############################################################################
##
InstallMethod( LoopsXMod, "for a crossed module and an element", true,
[ IsPreXMod, IsObject ], 0,
function( X0, a )
local M0, genM0, lenM0, P0, bdy0, act0, aut0, acta, elPa, p, c, m, rec0,
nPa, C0, G0, e1, e2, Pa, genPa, posPa, i, e, e2m, e1p, g, ngPa,
imda, j, pa, ma, bdy1, imact1, act1;
if not IsXMod( X0 ) then
Error( "X0 should be a crossed module" );
fi;
M0 := Source( X0 );
genM0 := GeneratorsOfGroup( M0 );
lenM0 := Length( genM0 );
P0 := Range( X0 );
if not ( a in P0 ) then
Error( "element a not in the range group P0" );
fi;
Info( InfoXMod, 3, "M0: ", Elements(M0) );
Info( InfoXMod, 3, List( Elements(M0), Order ) );
Info( InfoXMod, 3, "P0: ", Elements(P0) );
Info( InfoXMod, 3, List( Elements(P0), Order ) );
bdy0 := Boundary( X0 );
act0 := XModAction( X0 );
aut0 := Range( act0 );
acta := ImageElm( act0, a );
elPa := [ ];
for p in P0 do
c := Comm( p, a );
for m in M0 do
if ( ImageElm( bdy0, m ) = c ) then
Add( elPa, [p,m] );
fi;
od;
od;
nPa := Length( elPa );
Info( InfoXMod, 2, "elPa has length ", nPa );
Info( InfoXMod, 3, "with elements ", elPa );
C0 := Cat1GroupOfXMod( X0 );
G0 := Source( C0 );
if HasSemidirectProductInfo( G0 ) then
e1 := Embedding( G0, 1 );
e2 := Embedding( G0, 2 );
elif HasPreCat1GroupRecordOfPreXMod( X0 ) then
rec0 := PreCat1GroupRecordOfPreXMod( X0 );
e1 := rec0!.xmodRangeEmbeddingIsomorphism;
e2 := rec0!.xmodSourceEmbeddingIsomorphism;
else
Error( "no way of defining e1, e2" );
fi;
Pa := Subgroup( G0, [ ] );
genPa := [ ];
posPa := [ ];
for i in [1..nPa] do
e := elPa[i];
p := e[1];
m := e[2];
e2m := ImageElm( e2, m );
e1p := ImageElm( e1, p );
g := e1p * e2m;
if not ( g in Pa ) then
Add( genPa, g );
Add( posPa, i );
Pa := Subgroup( G0, genPa );
fi;
od;
Info( InfoXMod, 2, "Pa has size ", Size( Pa ) );
ngPa := Length( genPa );
## construct the boundary for X1
imda := ListWithIdenticalEntries( lenM0, 0 );
for j in [1..lenM0] do
m := genM0[j];
pa := ImageElm( bdy0, m );
ma := m^-1 * ImageElm( acta, m );
e2m := ImageElm( e2, ma );
e1p := ImageElm( e1, pa );
g := e1p * e2m;
if not ( g in Pa ) then
Error( "element g not in Pa" );
fi;
imda[j] := g;
od;
bdy1 := GroupHomomorphismByImages( M0, Pa, genM0, imda );
if ( bdy1 = fail ) then
Error( "bdy1 fails to be a homomorphism" );
fi;
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( "the boundary map:\n" );
Display( bdy1 );
fi;
## construct the action map for X1
imact1 := ListWithIdenticalEntries( ngPa, 0 );
for i in [1..ngPa] do
p := elPa[ posPa[i] ][1];
imact1[i] := ImageElm( act0, p );
od;
act1 := GroupHomomorphismByImages( Pa, aut0, genPa, imact1 );
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( "the action map:\n" );
Display( act1 );
fi;
## now we can construct the crossed module
return XModByBoundaryAndAction( bdy1, act1 );
end );
############################################################################
##
InstallMethod( AllLoopsXMod, "default method for a crossed module", true,
[ IsPreXMod ], 0,
function( X0 )
local bdy0, act0, M0, genM0, P0, genP0, igenP0, imbdy0, Q0,
conjP0, repsP0, eltsP0, nclP0, i, eclP0, relP0, c, p, genimbdy0,
len, class, j, r, k, q, pos, d, h, eqreps, numreps, a, X1, allX;
bdy0 := Boundary( X0 );
act0 := XModAction( X0 );
M0 := Source( X0 );
genM0 := GeneratorsOfGroup( M0 );
P0 := Range( X0 );
genP0 := GeneratorsOfGroup( P0 );
igenP0 := List( genP0, g -> ImageElm( act0, g ) );
imbdy0 := ImagesSource( bdy0 );
Q0 := P0/imbdy0;
conjP0 := ConjugacyClasses( P0 );
repsP0 := List( conjP0, Representative );
eltsP0 := List( conjP0, Elements );
nclP0 := Length( conjP0 );
if ( InfoLevel( InfoXMod ) > 1 ) then
Print( "#I Size(P0) = ", Size(P0), "\n" );
Print( "#I cokernel of boundary has IdGroup ", IdGroup(Q0), "\n" );
Print( "#I conjugacy class sizes, reps and orders:classes for P :-\n" );
Print( "#I ", List( conjP0, Size ), "\n" );
Print( "#I repsP0 = ", repsP0, "\n" );
fi;
if ( InfoLevel( InfoXMod ) > 2 ) then
Print( "\n#I orders and elements in conjugacy classes:\n" );
for i in [1..nclP0] do
Print( "#I ", i, " : ", Order(repsP0[i]), " > ", eltsP0[i], "\n" );
od;
fi;
eltsP0 := Elements( P0 );
eclP0 := ListWithIdenticalEntries( Size(P0), 1 );
relP0 := [1..nclP0];
for i in [2..nclP0] do
c := conjP0[i];
for p in c do
eclP0[ Position( eltsP0, p ) ] := i;
od;
od;
Info( InfoXMod, 3, "\neclP0 = ", eclP0 );
genimbdy0 := GeneratorsOfGroup( imbdy0 );
for i in [1..nclP0] do
if ( relP0[i] = i ) then
len := 1;
class := [ i ];
j := 0;
while ( j < len ) do
j := j+1;
c := class[j];
r := repsP0[c];
for k in imbdy0 do ##? genimbdy do
q := r*k;
pos := Position( eltsP0, q );
d := eclP0[pos]; ## the class of q
if ( d <> c ) then
Info( InfoXMod, 3, "[c,r,k,q,d] = ", [c,r,k,q,d] );
for h in [1..nclP0] do
if ( relP0[h] = d ) then
relP0[h] := c;
fi;
od;
Add( class, d );
Info( InfoXMod, 3, "class = ", class );
fi;
od;
od;
fi;
od;
eqreps := [ ];
for i in [1..nclP0] do
if ( i = relP0[i] ) then
Add( eqreps, repsP0[i] );
fi;
od;
numreps := Length( eqreps );
Info( InfoXMod, 2, "there are ", numreps,
" equivalence classes with representatives:" );
Info( InfoXMod, 2, eqreps );
## now run LoopsXMod on on the list eqreps
allX := ListWithIdenticalEntries( numreps, 0 );
for i in [1..numreps] do
a := eqreps[i];
X1 := LoopsXMod( X0, a );
Info( InfoXMod, 1, "LoopsXMod with a = ", a, ", ",
"IdGroup = ", IdGroup( X1 ) );
if ( InfoLevel( InfoXMod ) > 1 ) then
Display( X1 );
fi;
allX[i] := X1;
od;
return allX;
end );
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2026-03-28
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