Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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, the only other subgroups of $SL(2,5)$ are:

  1. Binary dihedral groups of order 12 and 20 (not nilpotent by
  2. The binary tetrahedral group of order 24 (by reading a bit in I figured out it is not the direct product of its Sylow subgroups, thus it's not nilpotent).

Is this correct?

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.