[GAP Forum] Basic question: picking out a certain subgroup of A_{31}

[Alexander Hulpke]

Dear Forum,

On Jan 26, 2011, at 1/26/11 2:35, William DeMeo wrote:
> How do I get ahold of a subgroup of A_{31} that is isomorphic to
> SL(5,2).  I know there's one in there, but I don't know how to get a
> handle on it.

Two ways:

a) The reason that SL_5(2) lies in S_31 is because it is the action on the nonzero vectors. We can just do this:

gap> G:=SL(5,2);
SL(5,2)
gap> vecs:=Filtered(Elements(GF(2)^5),x->not IsZero(x));;
gap> Length(vecs);
31
gap> K:=Action(G,vecs,OnRight);
Group([ (16,24)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31), 
  (1,2,4,8,16)(3,6,12,24,17)(5,10,20,9,18)(7,14,28,25,19)(11,22,13,26,21)(15,
    30,29,27,23) ])

b) (if building the action is too hard) Many of the small degree permutation representations of the simple groups are primitive and can be found in the primitive groups library:

gap> Size(SL(5,2));
9999360
gap> l:=AllPrimitiveGroups(NrMovedPoints,31,Size,9999360);
[ L(5, 2) ]
gap> K:=l[1];
L(5, 2)
 (of course if there were multiple candidates, or we were not sure, we'd have to confirm that it is the same group:

gap> IsomorphismTypeInfoFiniteSimpleGroup(SL(5,2));
rec( name := "A(4,2) = L(5,2) ", parameter := [ 5, 2 ], series := "L" )
gap> IsomorphismTypeInfoFiniteSimpleGroup(K);      
rec( name := "A(4,2) = L(5,2) ", parameter := [ 5, 2 ], series := "L" )

so we're safe.)

Best,

   Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
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