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FIX: Outer automorphisms of certain Chevalley groups.
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Instead of tables in ATLAS (that might have been misinterpreted), use the
explict presentations in Bray/Holt/Roney-Dougal.

This fixes #4318
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@hulpke
hulpke committed Mar 16, 2021
1 parent c7a0c6e commit 41e4737
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156 changes: 148 additions & 8 deletions grp/simple.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -887,9 +887,101 @@ local id;
return DataAboutSimpleGroup(id);
end);

# Tables for outer automorphisms from Bray/Holt/Roney-Dougal, p. 36/37
BindGlobal("OuterAutoSimplePres",function(class,n,q)
local p,e,f,rels,gp;
class:=UppercaseString(class);
p:=Factors(q)[1];
e:=LogInt(q,p);
if class="L" and n=2 then
f:=FreeGroup("d","f");
rels:=Concatenation("d",String(Gcd(q-1,2)),
"=f",String(e),"=[d,f]=1");
elif class="L" and n>=3 then
f:=FreeGroup("d","g","f");
rels:=Concatenation("d",String(Gcd(q-1,n)),
"=g2=f",String(e),"=[g,f]=1,d^g=D,d^f=d",String(p));
elif class="U" and n>=3 then
f:=FreeGroup("d","g","f");
rels:=Concatenation("d",String(Gcd(q+1,n)),
"=g2=1,f",String(e),"=g,d^g=D,d^f=d",String(p));
elif class="S" and n>=2 and (n<>4 or p<>2) then
f:=FreeGroup("d","f");
rels:=Concatenation("d",String(Gcd(q-1,2)),
"=f",String(e),"=[d,f]=1");
elif class="S" and n=4 and p=2 then
f:=FreeGroup("g","f");
rels:=Concatenation("g2=f,f",String(e),"=1");
elif (class="O" or class="O0" or class="OO") and n>=3 then
# special case of characteristic 2 -- copy S status
if p=2 then
f:=FreeGroup("d","f");
rels:=Concatenation("d",String(Gcd(q-1,2)),
"=f",String(e),"=[d,f]=1");
else
f:=FreeGroup("d","f");
rels:=Concatenation("d2=f",String(e),"=[d,f]=1");
fi;
elif class="O+" and n>=6 and IsEvenInt(n) and n<>8 and p=2 then
f:=FreeGroup("g","f");
rels:=Concatenation("g2=f",String(e),"=[g,f]=1");
elif class="O+" and n=8 and p=2 then
f:=FreeGroup("t","g","f");
rels:=Concatenation("t^3=g2=(gt)2=f",String(e),
"=[t,f]=[g,f]=1");
elif class="O-" and n>=4 and IsEvenInt(n) and p=2 then
f:=FreeGroup("g","f");
rels:=Concatenation("g2,f",String(e),"=g");
elif class="O+" and n=8 and IsOddInt(p) then
f:=FreeGroup("p","t","g","d","f");
rels:=Concatenation("p2=t3=g2=(gt)2=d2=1,d^t=p,p^dp,(dg)2=p,f",
String(e),"=[d,f]=[t,f]=[g,f]=1");
elif class="O+" and n>=12 and n mod 4=0 and IsOddInt(q) then
f:=FreeGroup("p","g","d","f");
rels:=Concatenation("p2=g2=d2=1,(dg)2=p,f",
String(e),"=[d,f]=[g,f]=1");
elif class="O+" and n>=6 and n mod 4=2 and q mod 4=1 then
f:=FreeGroup("p","g","d","f");
rels:=Concatenation("p2=g2=1,d2=p,d^g=D,f",
String(e),"=[g,f]=1,d^f=d",String(p));
elif class="O+" and n>=6 and n mod 4=2 and q mod 4=3 then
f:=FreeGroup("g","d","f");
rels:=Concatenation("g2=d2=[d,g]=f",
String(e),"=[g,f]=[d,f]=1");
elif class="O-" and n>=4 and (n mod 4=0 or q mod 4=1) and IsOddInt(q) then
f:=FreeGroup("g","d","f");
rels:=Concatenation("g2=d2=[d,g]=[d,f]=1,f",
String(e),"=g");
elif class="O-" and n>=4 and n mod 4=2 and q mod 4=3 then
f:=FreeGroup("p","g","d","f");
rels:=Concatenation("p2=g2=1,d2=p,d^g=D,f",
String(e),"=[g,f]=[d,f]=1");
else
return fail;
fi;
gp:=f/ParseRelators(f,rels);
SetReducedMultiplication(gp);
Size(gp);
return gp;
end);

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

shortname:=function(gp)
local s;
if IsCyclic(gp) then
return String(Size(gp));
elif Size(gp)<=31 or Size(gp) in [33..47] then
s:=StructureDescription(gp);
s:=Filtered(s,x->not x in "C ");
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!
Expand Down Expand Up @@ -1038,6 +1130,7 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
fi;

e:=[];
classical:=fail;
if id.series="Spor" then
nam:=id.name;
# deal wirth stupid names in identification
Expand Down Expand Up @@ -1076,16 +1169,19 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
e:=[[2,"2"]];
fi;
elif id.series="L" then
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);
else
e:=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);
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
Expand All @@ -1099,13 +1195,17 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
nam:=Concatenation("Sz(",String(par),")");
e:=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;
#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;
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);
Expand All @@ -1115,8 +1215,10 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
e:=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),expo/2,1);

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

elif id.series="F" then
nam:=Concatenation("F4(",String(par),")");
Expand Down Expand Up @@ -1150,8 +1252,9 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
e:=fail;
fi;

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

# get size of full outer automorphisms
Expand All @@ -1165,6 +1268,43 @@ local nam,e,EFactors,par,expo,prime,result,aut,i;
aut:=false;
fi;

if classical<>fail then
classaut:=OuterAutoSimplePres(classical[1],classical[2],classical[3]);
if classaut<>fail then
if IdGroup(classaut)=[4,2] then
# subgroup classes V4
i:=[[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" ] ];
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" ] ];
elif IdGroup(classaut)=[24,12] then
# subgroup classes S_4
i:=[ [ 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));
fi;
fi;
fi;


result:=rec(idSimple:=id,
tomName:=nam,
allExtensions:=e,
Expand Down
6 changes: 6 additions & 0 deletions tst/testbugfix/2021-03-15-chevallaut.tst
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Original file line number Diff line number Diff line change
@@ -0,0 +1,6 @@
# Outer automorphisms Chevalley groups miscomputed with triality, #4318
gap> G := PrimitiveGroup(119, 1);;
gap> S := SymmetricGroup(119);;
gap> N := Normalizer(S, G);;
gap> Index(N, G);
2

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