[GAP Forum] simple groups

[Joe Bohanon]

The smallest maximal subgroup of Ly has index  ~ 8.8 million.

Even asking GAP for the Sylow 2-subgroup of Ly is taking a long time on 
my faster processor.  With that group you have the distinct advantage of 
the maximal subgroup containing your Sylow 2-subgroup being one of the 
groups in the Atlas with a very small permutation representation.

A somewhat smaller, but still difficult, example is the O'Nan simple 
group.  Its smallest permutation rep has degree 122760 and GAP is also 
struggling to pick out a Sylow 2-subgroup.  As it turns out J1 is the 
largest maximal subgroup containing the Sylow 2-subgroup.  While GAP can 
pick out a Sylow 2-subgroup and get its rank in about 5 seconds, I get 
an almost instantaneous answer if I run:

gap> G:=Group(AtlasGenerators("ON",1,4).generators);;

gap> small:=SmallerDegreePermutationRepresentation(G);;

gap> NrMovedPoints(Image(small));
448

gap> SylowSubgroup(Image(small),2);
<permutation group of size 512 with 7 generators>

gap> RankPGroup(last);
3

I would say that any time you are trying to run a computation on a 
permutation group of largish degree, you should run 
SmallerDegreePermutationRepresentation and then use PreImage to get back 
to where you started if that's even necessary.  The maximal subgroups of 
almost all of the sporadic groups are in the Atlas, but when you call 
the function I did above to get J1 as a maximal subgroup of ON, it 
constructs it as a subgroup under the "1st" representation that the 
Atlas has.  Some groups don't have any permutation representation in the 
Atlas and many don't even have their maximal subgroups, so this doesn't 
work universally.

Joe

Jack Schmidt wrote:
>
> PerfectGroup by default creates a finitely presented group, and the 
> methods for calculating in finitely presented groups are slow.  You 
> can ask for a permutation group, and the calculation is very fast:
>
> gap> RankPGroup(SylowSubgroup(PerfectGroup(IsPermGroup,175560,1),2));
> 3
>
> The library of perfect groups only goes up to 10^6, but the GAP 
> package AtlasRep makes it easy to get the other groups:
>
> gap> LoadPackage("atlasrep");
> true
> gap> RankPGroup(SylowSubgroup(AtlasGroup("J1"),2));
> 3
> gap> RankPGroup(SylowSubgroup(AtlasGroup("J2"),2));
> 3
> gap> RankPGroup(SylowSubgroup(AtlasGroup("McL"),2));
> 3
> gap> Ly:=AtlasGroup("Ly");
> <matrix group of size 51765179004000000 with 2 generators>
>
> The Lyons group does not have a small permutation representation (I 
> believe the smallest is nearly 10 million points, but I don't have my 
> copy of the atlas at hand).  For this reason, it may be more difficult 
> to find the Sylow subgroup.  GAP will default try to find a 
> permutation representation first, which will probably fail since many 
> of its permutation representations are much larger than 10 million 
> points.
>
> However, you can also consider maximal subgroups of the Lyons group 
> that contain the Sylow 2-subgroup.  One of these is 2.A11, which I 
> believe is the double cover of A11.  I think it should be clear that 
> the rank of the Sylow 2-subgroup of A11 and 2.A11 are equal, and the 
> rank of A11 is found from:
> gap> RankPGroup(SylowSubgroup(AlternatingGroup(11),2));
> 3
>
> Alternatively one can ask atlasrep again:
> gap> RankPGroup(SylowSubgroup(AtlasGroup("2.A11"),2));
> 3
>
>
> I think the notation with the lower . is supposed to be ambiguous, so 
> one could entertain the possibility that 2 x A11 is a maximal subgroup 
> of the Lyons group, and then the rank would be 4.  I don't think this 
> is right, but I feel safer saying the rank is either 3 or 4.
>
>
> Alex Trofimuk wrote:
>>
>> -- Dear Gap Forum,
>> Alex Trofimuk asked:
>> Using function PerfectGroup(), I defined simple groups Janko J1, J2. 
>> But I can not find rank of their Sylow 2-subgroups. I used function 
>> RankPGroup().  Probably, my computer has weak power.  Help me, 
>> please, to find it. How to define groups Janko J3, J4 and sporadic 
>> simple groups Mc, Ly in system Gap. Is it possible to calculate rank 
>> of their Sylow 2-subgroups?
>>
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