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

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merged 7 commits into from
Apr 20, 2021
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Merged

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

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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
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272 changes: 84 additions & 188 deletions grp/simple.gi
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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
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