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,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hul...@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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