Description
Take this input group:
G := Group(Z(5)^0 * [
[ [ 0, 2, 0, 2 ], [ 1, 4, 4, 4 ],
[ 0, 4, 0, 0 ], [ 3, 2, 0, 0 ] ],
[ [ 0, 2, 0, 2 ], [ 4, 4, 1, 4 ],
[ 0, 1, 0, 0 ], [ 3, 3, 0, 0 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 4, 0, 0 ],
[ 0, 0, 4, 0 ], [ 0, 0, 0, 1 ] ],
[ [ 2, 0, 0, 0 ], [ 0, 2, 0, 0 ],
[ 0, 0, 2, 0 ], [ 0, 0, 0, 2 ] ],
[ [ 0, 0, 0, 1 ], [ 0, 2, 0, 0 ],
[ 0, 0, 1, 0 ], [ 2, 0, 0, 0 ] ]
]);
Trying RecognizeGroup(G) often (= for many RNG seeds) prints one or more messages of the form
#I Strange, found less direct factors than expected!
which already indicates something is amiss.
But it sometimes also runs into an outright error:
gap> G:=Group(Z(5)^0*[[[0,2,0,2],[1,4,4,4],[0,4,0,0],[3,2,0,0]],[[0,2,0,2],[4,4,1,4],[0,1,0,0],[3,3,0,0]],[[1,0,0,0],[0,4,0,0],[0,0,4,0],[0,0,0,1]],[[2,0,0,0],[0,2,0,0],[0,0,2,0],[0,0,0,2]],[[0,0,0,1],[0,2,0,0],[0,0,1,0],[2,0,0,0]]]);
gap> seed:=115;; Reset(GlobalMersenneTwister,seed);; Reset(GlobalRandomSource,seed);; RecognizeGroup(G);
Error, ^ cannot be used here to compute roots (use `RootInt' instead?) at GAPROOT/lib/arith.gi:623 called from
z ^ ((- log) * ri!.gcd.coeff1 / ri!.gcd.gcd)
at GAPROOT/pkg/recog/gap/projective/classicalnatural.gi:3068 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:711 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:733 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:711 called from
SLPforElement( rifac, gg ) at GAPROOT/pkg/recog/gap/base/recognition.gi:733 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:711 called from
SLPforElement( riker, n ) at GAPROOT/pkg/recog/gap/base/recognition.gi:753 called from
slpforelement( ri )( ri, x ) at GAPROOT/pkg/recog/gap/base/recognition.gi:711 called from
SLPforElement( ImageRecogNode( ri ), ImageElm( Homom( ri ), x!.el ) ) at GAPROOT/pkg/recog/gap/base/kernel.gi:46 called from
GenerateRandomKernelElementsAndOptionallyVerifyThem( ri, n, false ) at GAPROOT/pkg/recog/gap/base/kernel.gi:91 called from
CallFuncList( findgensNmeth( ri ).method, Concatenation( [ ri ], findgensNmeth( ri ).args ) ) at GAPROOT/pkg/recog/gap/base/recognition.gi:590 called from
RecogniseGeneric( G, FindHomDbMatrix, "", rec( ) ) at GAPROOT/pkg/recog/gap/base/recognition.gi:112 called from
RecogniseGroup( G ) at GAPROOT/ClassicalMaximals/tst/utils.g:8 called from
CheckSize( G ); at stream:5 called from
TestOrthogonalNormalizerInSL( 1, 4, 5 ); at *stdin*:5 called from
<function "unknown">( <arguments> )
called from read-eval loop at *errin*:12
brk>
This is the code in question:
SLPforElementFuncsProjective.PSL2 := function(ri,x)
local det,log,slp,y,z,pos,s;
ri!
5B84
.fakegens.count := ri!.fakegens.count + 1;
if ri!.fakegens.count > 1000 then
ri!.fakegens := RECOG.InitSLfake(ri!.field,2);
ri!.fakegens.count := 0;
fi;
y := ri!.nicebas * x * ri!.nicebasi;
det := DeterminantMat(y);
if not IsOne(det) then
z := PrimitiveRoot(ri!.field);
log := LogFFE(det,z);
y := y * z^(-log*ri!.gcd.coeff1/ri!.gcd.gcd); # <-- error happens here
fi;
brk>
So at this point det = z = Z(5), so log = 1, and ri!.gcd = rec( coeff1 := 1, coeff2 := 0, coeff3 := -2, coeff4 := 1, gcd := 2 ), so the exponent evaluates to -1/2.
It looks as if the code is trying to normalize the determinant of that matrix to one, by scaling the matrix with a square root of the inverse of that determinant -- but of course that doesn't exist in GF(5).
So the input matrix is not actually "in" PSL(2,5) (or rather, in SL(2,5)). Indeed, let's inspect some more on the break loop:
brk> Grp(ri);
<matrix group with 24 generators>
brk> Grp(ri).1;
[ [ 0*Z(5), Z(5)^2 ], [ Z(5)^0, Z(5)^2 ] ]
brk> Size(Grp(ri));
240
brk> Size(SL(2,5));
120
So the matrices generated GL(2,5), not SL(2,5). So this looks like a case of "mistaken identity": The code drew a wrong conclusion, it thinks it has (P)SL(2,5) but this is wrong.