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

[William DeMeo]

Thank you, Dr. Hulpke, for your exceedingly clear explanation.  Option
(a) is simple enough, but option (b) is interesting too.

By the way, I asked the question because I wanted to see if XGAP could
actually draw Feit's example of an M_7 lattice as an upper interval in
the subgroup lattice of A_{31}, and I've now got all the vertices
drawn except the six copies of SL(5,2).  If interested, read on,
otherwise, feel free to ignore...

I believe Feit takes a Sylow-31 subgroup P of A_{31}, then looks at
the normalizer N = N(P) and notices that |N| = 31.15 and that N
contains a subgroup H of order 31.5.  Then he asserts (with little
justification) that there are two non-conjugate copies of SL(5,2)
above H, call them K1 and K2.  If you conjugate each of these by the
(three) coset representatives of N/H, you get six of the coatoms of
the M_7 interval.  The seventh coatom is N.

Now, thanks to your help, at least I can find a SL(5,2) subgroup of
A_{31}. Unfortunately, it's not one of the SL(5,2)'s above H. (It's K
in the attached diagram.)  Anyway, I'm sure I can figure it out from
here.  Thanks again for your help!!

-William


On Wed, Jan 26, 2011 at 11:53 AM, Alexander Hulpke
<hulpke at math.colostate.edu> wrote:
>
>
> 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: hulpke at math.colostate.edu, Phone: ++1-970-4914288
> http://www.math.colostate.edu/~hulpke
>
>
>
>