[GAP Forum] simple groups

[Jack Schmidt]

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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