Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

Collection of small fixes and improvements that do not merit an own PR each #4411

Merged
merged 7 commits into from
Apr 20, 2021
Merged

Collection of small fixes and improvements that do not merit an own PR each #4411

Show file tree
Hide file tree
Changes from 6 commits
Commits
Show all changes
7 commits
Select commit Hold shift + click to select a range
ac5c7b0
ENHANCE: Criterion for using radical automorphisms
hulpke Mar 31, 2021
374b566
ENHANCE: Clean away `EFactors' function.
hulpke Apr 6, 2021
c50bf51
ENHANCE: Transfer solvability information onto extension.
hulpke Apr 6, 2021
59ccd67
ENHANCE: RankAction tests transitivity
hulpke Apr 12, 2021
93568a8
FIX: MTC will do better work in deleting superfluous generators.
hulpke Apr 16, 2021
b78dc2e
ENHANCE: IntermediateSubgroups performance
hulpke Apr 20, 2021
ed322bb
Update lib/ghomfp.gi
hulpke Apr 20, 2021
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
272 changes: 84 additions & 188 deletions grp/simple.gi
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -956,150 +956,46 @@ end);

InstallOtherMethod(DataAboutSimpleGroup,true,[IsRecord],0,
function(id)
local
nam,e,EFactors,par,expo,prime,result,aut,i,classical,classaut,shortname;
local nam,e,efactors,par,expo,prime,result,aut,i,classical,classaut,shortname,
multElabel;

shortname:=function(gp)
local s;
if IsCyclic(gp) then
return String(Size(gp));
elif IdGroup(gp)=[4,2] then
return "2^2";
elif IdGroup(gp)=[6,1] then
return "3.2";
elif IdGroup(gp)=[8,3] then
return "2^2.2";
elif IdGroup(gp)=[9,2] then
return "3^2";
elif IdGroup(gp)=[18,3] then
return "3^2.2";
elif Size(gp)<=31 or Size(gp) in [33..47] then
s:=StructureDescription(gp);
s:=Filtered(s,x->not x in "C ");
if Length(s)>3 and s{[Length(s)-2..Length(s)]}="xS3" then
s:=Concatenation(s{[1..Length(s)-3]},".3.2");
fi;
return s;
else
Error("name not yet found");
fi;
end;

#note: Extensions are up to isomorphism, not with embedding questions. Thus
#no distinction of ' extensions!

# possible automorphism extensions for Chevalley groups
EFactors:=function(d,f,g)
local dd,df,dg,r,myprod,gal,i,s,j,ddn;

if g>2 then
# so triality is involved

# do a few basic cases first
if d=1 and f=1 and g=6 then
# subgroup classes S_3
dd:=[ [ 1, "1" ], [ 2, "2" ], [ 3, "3" ], [ 6, "3.2" ] ];
return dd;
elif d=1 and f=2 and g=6 then
# subgroup classes 2\times S_3 (since S3 cannot act on C2)
dd:=[ [ 1, "1" ], [ 2, "2_1" ],[ 2, "2_2" ], [ 2, "2_3" ],
[ 3, "3" ], [ 4, "2x2" ], [ 6, "6" ], [6,"S3_1"],
[ 6, "S3_2" ], [ 12, "D12" ] ];
elif d=4 and f=1 and g=6 then
# subgroup classes S_4
dd:=[ [ 1, "1" ], [ 2, "2_1" ],[ 2, "2_2" ], [ 3, "3" ],
[ 4, "4" ], [ 4, "(2^2)_{111}" ], [4,"(2^2)_{122}"],
[ 6, "3.2" ], [ 8, "D8" ], [ 12, "A4" ],
[ 24, "S4" ] ];
return dd;
else
Info(InfoWarning, "DataAboutSimpleGroup: mixed triality not yet done");
return fail;
fi;
fi;

if f>1 then
dd:=PrimitiveElement(GF(prime^f))^((prime^f-1)/d);
dd:=List([0..d-1],x->dd^x); # Powers;
gal:=GaloisGroup(GF(prime^f));
gal:=SmallGeneratingSet(gal);
if Length(gal)>1 then Error("not small generators");fi;
gal:=gal[1];
gal:=List([0..f-1],x->Permutation(gal^x,dd));
fi;

myprod:=function(c1,c2)
local d2,g1,f1;
d2:=c2[1];
if c1[3]>0 and d2>0 then
d2:=((d2+1)^gal[c1[3]+1])-1;
fi;

if c1[2]=1 then
# g-action
d2:=-d2 mod d;
fi;

if g=1 then
g1:=0;
else
g1:=(c1[2]+c2[2]) mod g;
multElabel:=function()
local a,b,j;
for a in e do
b:=Filtered(e,x->x[2]=a[2]);
if Length(b)>1 then
for j in [1..Length(b)] do
b[j][2]:=Concatenation(b[j][2],"_",String(j));
od;
fi;

if f=1 then
f1:=0;
else
f1:=(c1[3]+c2[3]) mod f;
fi;
f1:=[(c1[1]+d2) mod d,g1,f1];
#Print(c1,"*",c2,"=",f1,"\n");
return f1;

end;

r:=[];
if Number([d,g,f],i->i>1)>1 then
# form group
dd:=Cartesian([0..d-1],[0..g-1],[0..f-1]);
dd:=List(dd,x->List(dd,y->Position(dd,myprod(y,x))));
dd:=List(dd,PermList);
dd:=Group(dd);
if Size(dd)<>d*f*g then Error("wrong automorphism order");fi;
dd:=List(ConjugacyClassesSubgroups(dd),Representative);
ddn:=[];
for i in dd do
if IsCyclic(i) then
Add(ddn,String(Size(i)));
else
r:="";
s:=Reversed(ElementaryAbelianSeriesLargeSteps(i));
for j in [2..Length(s)] do
if Length(r)>0 then
Add(r,'.');
fi;
j:=AbelianInvariants(s[j]/s[j-1]);
Append(r,String(j[1]));
if Length(j)>1 then
Add(r,'^');
Append(r,String(Length(j)));
fi;
od;
Add(ddn,r);
fi;
od;
r:=[];
for i in [1..Length(dd)] do
if Number(ddn,x->x=ddn[i])=1 then
Add(r,[Size(dd[i]),ddn[i]]);
else
Add(r,[Size(dd[i]),Concatenation(ddn[i],"_",
String(Number(ddn{[1..i]},x->x=ddn[i])))]);
fi;
od;

return r;
fi;

if d>1 then
dd:=DivisorsInt(d);
r:=List(dd,i->[i,String(i)]);
fi;
if f>1 then
df:=DivisorsInt(f);
r:=List(df,i->[i,String(i)]);
fi;
if g>1 then
dg:=DivisorsInt(g);
r:=List(dg,i->[i,String(i)]);
fi;
return Filtered(r,i->i[1]>1);
od;
Sort(e);
end;

# fix O5 to SP4
Expand All @@ -1118,7 +1014,9 @@ nam,e,EFactors,par,expo,prime,result,aut,i,classical,classaut,shortname;
fi;
fi;

e:=[];
efactors:=fail;
e:=fail;

classical:=fail;
if id.series="Spor" then
nam:=id.name;
Expand Down Expand Up @@ -1149,6 +1047,8 @@ nam,e,EFactors,par,expo,prime,result,aut,i,classical,classaut,shortname;
if nam in ["M12","M22","HS","McL","He","Fi22","F3+","HN","Suz","ON",
"J2","J3"] then
e:=[[2,"2"]];
else
e:=[];
fi;
elif id.series="A" then
nam:=Concatenation("A",String(par));
Expand All @@ -1161,76 +1061,74 @@ nam,e,EFactors,par,expo,prime,result,aut,i,classical,classaut,shortname;
classical:=["L",par[1],par[2]];
nam:=Concatenation("L",String(par[1]),"(",String(par[2]),")");
if par[1]=2 then
e:=EFactors(Gcd(2,par[2]-1),expo,1);
efactors:=[Gcd(2,par[2]-1),expo,1];
else
e:=EFactors(Gcd(par[1],par[2]-1),expo,2);
efactors:=[Gcd(par[1],par[2]-1),expo,2];
fi;
elif id.series="2A" then
classical:=["U",par[1]+1,par[2]];
nam:=Concatenation("U",String(par[1]+1),"(",String(par[2]),")");
e:=EFactors(Gcd(par[1]+1,par[2]+1),2*expo,1);
efactors:=[Gcd(par[1]+1,par[2]+1),2*expo,1];
elif id.series="B" then
classical:=["O",2*par[1]+1,par[2]];
nam:=Concatenation("O",String(2*par[1]+1),"(",String(par[2]),")");
if par[1]=2 and par[2]=3 then
nam:="U4(2)"; # library name
fi;
if par[1]=2 and prime=2 then
e:=EFactors(Gcd(2,par[2]-1),expo,2);
efactors:=[Gcd(2,par[2]-1),expo,2];
else
e:=EFactors(Gcd(2,par[2]-1),expo,1);
efactors:=[Gcd(2,par[2]-1),expo,1];
fi;
elif id.series="2B" then
nam:=Concatenation("Sz(",String(par),")");
e:=EFactors(1,expo,1);
efactors:=[1,expo,1];
elif id.series="C" then
classical:=["S",2*par[1],par[2]];
nam:=Concatenation("S",String(par[1]*2),"(",String(par[2]),")");
#if par[1]=2 and prime=2 then
# e:=EFactors(Gcd(2,par[2]-1),expo,2);
#else
# e:=EFactors(Gcd(2,par[2]-1),expo,1);
#fi;
# efactors finds wrong structure
e:=fail;
if par[1]=2 and prime=2 then
efactors:=[Gcd(2,par[2]-1),expo,2];
else
efactors:=[Gcd(2,par[2]-1),expo,1];
fi;
elif id.series="D" then
classical:=["O+",2*par[1],par[2]];
nam:=Concatenation("O",String(par[1]*2),"+(",String(par[2]),")");
if par[1]=4 then
e:=EFactors(Gcd(2,par[2]-1)^2,expo,6);
efactors:=[Gcd(2,par[2]-1)^2,expo,6];
elif IsEvenInt(par[1]) then
e:=EFactors(Gcd(2,par[2]-1)^2,expo,2);
efactors:=[Gcd(2,par[2]-1)^2,expo,2];
else
e:=EFactors(Gcd(4,par[2]^par[1]-1),expo,2);
efactors:=[Gcd(4,par[2]^par[1]-1),expo,2];
fi;
elif id.series="2D" then
classical:=["O-",2*par[1],par[2]];
nam:=Concatenation("O",String(par[1]*2),"-(",String(par[2]),")");

e:=EFactors(Gcd(4,par[2]^par[1]+1),2*expo,1);
efactors:=[Gcd(4,par[2]^par[1]+1),2*expo,1];

elif id.series="F" then
nam:=Concatenation("F4(",String(par),")");
if prime=2 then
e:=EFactors(1,expo,2);
efactors:=[1,expo,2];
else
e:=EFactors(1,expo,1);
efactors:=[1,expo,1];
fi;

elif id.series="G" then
nam:=Concatenation("G2(",String(par),")");
if prime=3 then
e:=EFactors(1,expo,2);
efactors:=[1,expo,2];
else
e:=EFactors(1,expo,1);
efactors:=[1,expo,1];
fi;

elif id.series="3D" then
nam:=Concatenation("3D4(",String(par),")");
e:=EFactors(1,3*expo,1);
efactors:=[1,3*expo,1];
elif id.series="2G" then
nam:=Concatenation("R(",String(par),")");
e:=EFactors(1,expo,1);
efactors:=[1,expo,1];
elif id.series="2F" and id.parameter=2 then
# special case for tits' group before sorting out further 2F4's
nam:="2F4(2)'";
Expand All @@ -1241,66 +1139,64 @@ nam,e,EFactors,par,expo,prime,result,aut,i,classical,classaut,shortname;
e:=fail;
fi;

# get extensions
if IsList(e) then
# kill trivial extension if given
e:=Filtered(e,x->x[1]>1);

# get size of full outer automorphisms
aut:=[1,nam];
for i in e do
if i[1]>aut[1] then
aut:=i;
fi;
od;
else
aut:=false;
fi;

aut:=fail;
classaut:=fail;
if classical<>fail then
classaut:=OuterAutoSimplePres(classical[1],classical[2],classical[3]);
if classaut<>fail then
if efactors<>fail and Size(classaut)<>Product(efactors) then
Error("outer automorphism efactor fail");
fi;
if IdGroup(classaut)=[4,2] then
# subgroup classes V4
i:=[[2,"2_1"],[2,"2_2"],[2,"2_3"],[4,"2^2"]];
e:=[[2,"2_1"],[2,"2_2"],[2,"2_3"],[4,"2^2"]];
elif IdGroup(classaut)=[6,1] then
# subgroup classes S_3
i:=[ [ 2, "2" ], [ 3, "3" ], [ 6, "3.2" ] ];
e:=[ [ 2, "2" ], [ 3, "3" ], [ 6, "3.2" ] ];
elif IdGroup(classaut)=[12,4] then
# subgroup classes 2\times S_3 (since S3 cannot act on C2)
i:=[ [ 2, "2_1" ],[ 2, "2_2" ], [ 2, "2_3" ],
[ 3, "3" ], [ 4, "2x2" ], [ 6, "6" ], [6,"S3_1"],
[ 6, "S3_2" ], [ 12, "D12" ] ];
e:=[ [ 2, "2_1" ], [ 2, "2_2" ], [ 2, "2_3" ], [ 3, "3" ],
[ 4, "2^2" ], [ 6, "3.2_1" ], [ 6, "3.2_2" ], [ 6, "6" ],
[ 12, "3.2^2" ] ];
elif IdGroup(classaut)=[24,12] then
# subgroup classes S_4
i:=[ [ 2, "2_1" ],[ 2, "2_2" ], [ 3, "3" ],
e:=[ [ 2, "2_1" ],[ 2, "2_2" ], [ 3, "3" ],
[ 4, "4" ], [ 4, "(2^2)_{111}" ], [4,"(2^2)_{122}"],
[ 6, "3.2" ], [ 8, "D8" ], [ 12, "A4" ],
[ 24, "S4" ] ];
else
i:=List(ConjugacyClassesSubgroups(classaut),Representative);
i:=Filtered(i,x->Size(x)>1);
i:=List(i,x->[Size(x),shortname(x)]);
fi;
if e<>fail then
if Collected(List(i,x->x[1]))<>Collected(List(e,x->x[1])) then
Error("outer automorphism disagreement error");
fi;
else
e:=i;
aut:=First(e,x->x[1]=Size(classaut));
e:=List(ConjugacyClassesSubgroups(classaut),Representative);
e:=Filtered(e,x->Size(x)>1);
e:=List(e,x->[Size(x),shortname(x)]);
multElabel();
fi;
fi;
elif e=fail and efactors<>fail then
classaut:=AbelianGroup(efactors);
e:=List(ConjugacyClassesSubgroups(classaut),Representative);
e:=Filtered(e,x->Size(x)>1);
e:=List(e,x->[Size(x),shortname(x)]);
multElabel();
fi;
if e=fail then Error("eh?");fi;

if aut=fail then
if Length(e)=0 then
aut:=[1,"1"];
else
aut:=Maximum(e);
fi;
fi;

result:=rec(idSimple:=id,
tomName:=nam,
allExtensions:=e,
fullAutGroup:=aut,
classicalId:=ClassicalIsomorphismTypeFiniteSimpleGroup(id));
if aut<>false then
result.fullAutGroup:=aut;
if classaut<>fail then
result.outerAutomorphismGroup:=classaut;
fi;
if efactors<>fail then result.efactors:=efactors;fi;

return result;
end);
Expand Down
Loading