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What are all nilpotent subgroups of the binary icosahedral group $SL(2,5)$, up to isomorphism? I need this for some sanity check so I'm currently more interested in what the result is than in how to get to it. But it would be nice to see the way too.

My attempt:

The nilpotent subgroups are:

  1. Cyclic Subgroups: Using a short Python program, I found out that each cyclic subgroup is of order 1,2,3,4,5,6 or 10.

  2. $Q_8$, the Lipschitz units (by Wikipedia).

By http://en.wikipedia.org/wiki/Binary_icosahedral_group, the only other subgroups of $SL(2,5)$ are:

  1. Binary dihedral groups of order 12 and 20 (not nilpotent by http://groupprops.subwiki.org/wiki/Dicyclic_group).
  2. The binary tetrahedral group of order 24 (by reading a bit in http://en.wikipedia.org/wiki/Binary_tetrahedral_group I figured out it is not the direct product of its Sylow subgroups, thus it's not nilpotent).

Is this correct?

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1 Answer

up vote 4 down vote accepted

Yes, this is correct. You can also ask such question to GAP. For example:

gap> g:=SL(2,5);;
gap> hs:=List(ConjugacyClassesSubgroups(g),Representative);;
gap> List(Filtered(hs,IsNilpotent),StructureDescription);
[ "1", "C2", "C3", "C4", "C5", "C6", "Q8", "C10" ]
gap> List(Filtered(hs,h -> not IsNilpotent(h)),StructureDescription);
[ "C3 : C4", "C5 : C4", "SL(2,3)", "SL(2,5)" ]

Here 1, C2, C3, C4, C5, C6, C10 are your cyclic groups, and Q8 is the Lipschitz units. C3 : C4 and C5 : C4 are dicyclic groups of orders 12 and 20, SL(2,3) is the binary tetrahedral group, and SL(2,5) is the binary icosahedral group.

If you prefer python programming, you may also want to look at Sage, a mathematics program written in python that includes interfaces to GAP and other popular mathematics programs.

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