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#############################################################################
##
#W extend.gi Bettina Eick
##
#W Enlarge a base pcgs by normalizing elements.
##
#############################################################################
##
#F SubsWordPlus( w, gens, invs, id ) . . . . . . . .use inverses and identity
##
BindGlobal( "SubsWordPlus", function( w, gens, invs, id )
local g, v;
g := id;
for v in w do
if v[2] = 1 then
g := g * gens[v[1]];
elif v[2] = -1 then
g := g * invs[v[1]];
elif v[2] > 1 then
g := g * gens[v[1]] ^ v[2];
elif v[2] < -1 then
g := g * invs[v[1]] ^ -v[2];
fi;
od;
return g;
end );
#############################################################################
##
#F SubsAndInvertDefn( w, defns ) . . . . . . . . . . . .substitute and invert
##
BindGlobal( "SubsAndInvertDefn", function( w, defns )
local v, l;
v := [];
for l in w do
Add( v, [defns[l[1]], -l[2]] );
od;
return Reversed( v );
end );
#############################################################################
##
#F TransWord( j, trels ) . . . . . . . . . . . . . determine transversal word
##
BindGlobal( "TransWord", function( j, trels )
local l, g, s, p, t, w;
l := Product( trels );
j := j - 1;
w := [];
for s in Reversed( [1..Length( trels )] ) do
p := trels[s];
l := l/p;
t := QuoInt( j, l );
j := RemInt( j, l );
if t > 0 then Add( w, [s, t] ); fi;
od;
return Reversed( w );
end );
#############################################################################
##
#F EnlargeOrbit( orbit, g, p, op ) . . . . enlarge orbit by p images under g
##
BindGlobal( "EnlargeOrbit", function( orbit, g, p, op )
local l, s, k, t, h;
l := Length( orbit );
orbit[p*l] := true;
s := 0;
for k in [ 1 .. p - 1 ] do
t := s + l;
for h in [ 1 .. l ] do
orbit[h+t] := op( orbit[h+s], g );
od;
s := t;
od;
end );
#############################################################################
##
#F SmallOrbitPoint( pcgs, g )
##
BindGlobal( "SmallOrbitPoint", function( pcgs, g )
local b;
repeat
b := Random(pcgs.acton);
until pcgs.oper( b, g ) <> b;
return b;
end );
#############################################################################
##
#F ExtendedBasePcgs( pcgs, g, d ) . . . . . . . . . . . . extend a base pcgs
##
## g normalizes <pcgs> and we compute a new pcgs for <pcgs, g>.
##
BindGlobal( "ExtendedBasePcgs", function( pcgs, g, d )
local h, e, i, o, b, m, c, l, w, j, k;
# change in place - but unbind not updated information
Unbind(pcgs.pcgs);
Unbind(pcgs.rels);
# set up
h := g;
e := ShallowCopy( d );
i := 0;
# loop over base and divide off
while not pcgs.trivl( h ) do
i := i + 1;
# take base point (if necessary, add new base point)
if i > Length( pcgs.orbit ) then
b := SmallOrbitPoint( pcgs, g );
Add( pcgs.orbit, [b] );
Add( pcgs.trans, [] );
Add( pcgs.defns, [] );
Add( pcgs.trels, [] );
else
b := pcgs.orbit[i][1];
fi;
# compute the relative orbit length of h
m := 1;
c := pcgs.oper( b, h );
while not c in pcgs.orbit[i] do
m := m + 1;
c := pcgs.oper( c, h );
od;
# enlarge pcgs, if necessary
if m > 1 then
#Print(" enlarge basic orbit ",i," by ",m," copies \n");
Add( pcgs.trans[i], h );
Add( pcgs.defns[i], e );
Add( pcgs.trels[i], m );
EnlargeOrbit( pcgs.orbit[i], h, m, pcgs.oper );
Add( pcgs.pcref, [i, Length(pcgs.trans[i])] );
fi;
# divide off
j := Position( pcgs.orbit[i], c );
if j > 1 then
w := TransWord( j, pcgs.trels[i] );
h := h^m * SubsWord( w, pcgs.trans[i] )^-1;
e := [[e,m], SubsAndInvertDefn( w, pcgs.defns[i] ) ];
else
h := h^m;
e := [e,m];
fi;
od;
end );
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