The binary polyhedral groups are finite subgroups of the quaternions corresponding (via McKay's ADE classification) to the $E$ series of affine Dynkin diagrams. They are also the lifts to $\mathrm{Spin}(3) \cong \mathrm{Sp}(1)$ of the groups of rotational symmetries of the platonic solids: the tetrahedron, the cube/octahedron and the dodecahedron/icosahedron.

The normal subgroups of these groups are well-known: they are even listed in their respective Wikipedia entries, but without a reference.

I need this result in a paper that I am writing and need a slightly more authoritative reference than the wikipedia page. Can someone point me in the right direction?

  • $\begingroup$ I should perhaps mention that no reference is needed for the group of type $E_8$, since its quotient by the centre is simple. I'm particularly interested in the groups of type $E_6$ and $E_7$. Although one can do the calculation (normal closures of conjugacy classes,...) I was hoping to be able to point to a reference. $\endgroup$ Jun 16, 2010 at 0:19
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    $\begingroup$ All these groups have a unique minimal normal subgroup (their center), so you can basically do this in their quotients (A_4, S_4, and A_5). But I guess you want a reference for their having a unique minimal normal subgroup? $\endgroup$
    – Steve D
    Jun 16, 2010 at 0:44
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    $\begingroup$ I didn't see a clear reference. M. Hall's Theory of Groups 16.10 and ams.org/mathscinet-getitem?mr=1298751 have enough information to calculate the normal subgroups easily (mentioning the unique minimal normal subgroup, displaying the character table). Both are more interested in their character theory, both are conversational, and neither clearly states that the groups have a unique chief series. Perhaps you could say "further properties of these groups are detailed in... and can be calculated by hand as needed." $\endgroup$ Jun 16, 2010 at 1:17
  • $\begingroup$ Thank you both for your comments. Hall's book is the only one I have handy and it looks reasonable. $\endgroup$ Jun 16, 2010 at 1:35
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    $\begingroup$ Springer's "Invariant Theory" (1977) has a brief historical survey and mentions 3 references, none of which I have close at hand: 1) Klein's "Lectures on the icosahedron", 2) Coxeter, "Regular complex polytopes", 3) Du Val, "Homographies, quaternions, and rotations". These are of course two-dimensional complex reflection groups and MathSciNet mentions 2 new books: Michel Broué, "Introduction to complex reflection groups and their braid groups" LNM 1988 and "Unitary reflection groups" by Lehrer and Taylor. I'd be surprised if it weren't there. $\endgroup$ Jun 16, 2010 at 2:31

1 Answer 1


You could do the computation with a short GAP program, and then cite GAP. It is substantially more authoritative than Wikipedia, though perhaps less popularly accepted than refereed papers. You might get complaints from a referee if he or she is especially square.

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    $\begingroup$ For that matter, you can cite the online ATLAS. $\endgroup$ Jun 16, 2010 at 4:42

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