added pointer to:

- Satoshi Tomoda, Peter Zvengrowski, Section 4.3 of:
*Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 (arXiv:0904.1876)

for discussion of the group cohomology

]]>added definition of $2I$ in terms of unit quaterions and relation to the 24-cell (here)

(entry needs cleaning up/streamlining, but not now)

]]>added the statement (here) that the binary icosahedral group is perfect (currently without proof or pointer to the literature)

]]>Thanks!

I have copied this also to *Platonic Solids – Symmetry groups*.

I added a stubby definition section.

]]>Right. Now does anyone have the time to write this out in the entry?

(I really don’t have any spare time right now. sorry.)

]]>Sure, sure. All I’m saying (looking at #5) is that there should be no ambiguity under any reasonable interpretation: you could just as well refer to combinatorial, not metric structure, and still get it right.

]]>Sorry, is there an actual question as to what the definition is? What Zoran is referring to is: consider the standard embedding of the Platonic solid into $\mathbb{R}^3$. Then a symmetry is an isometry of $\mathbb{R}^3$ that fixes the image of this embedding.

I suppose that’s pretty much the archetypical case of what people back then and laymen still today understand as a symmetry.

]]>Wouldn’t it be true that the automorphism group of the poset of facets (vertices, edges, faces) is the same as the isometry group of the regular Platonic solid? That would be my immediate understanding of what “symmetry” means here.

]]>I think for the *Idea*-section the word “symmetry” is just fine. What you should do is open a *Definition*-section and state the precise definition there!

You use ambiguous wording “symmetry” at icosahedral group where you apparently mean “isometry”. Is there a reason ? Should we change to isometry ?

]]>Thanks for this! I have added the pointers on discrete 2-group enumeration to the References at *2-group*

I added a few more points, including the exceptional isomorphisms to Lie-type groups over finite fields.

]]>I wonder if a crossed module presenting $\mathcal{I}$ was already found in the Bangor school’s work on enumerating small examples? Having a look around, I think it might be out of range, given that $|2I| = 120$, and so $|Mor(\mathcal{I})| = 24\times 120 = 2880$ (for others reading, this is not meant to be obvious: it’s a theorem in work of Epa-Ganter). The best I could find were this paper of Alp-Wensley (originally students at Bangor) on their GAP package with all isomorphism classes up to size 30, the latest GAP page for their package which says they have iso classes up to size 70, and this preprint of Ellis-van Luyen going up to 255.

But perhaps someone already found this exceptional example in one of those groups…

]]>added to *icosahedral group* discussion of the distinction of definitions as one moves up the Whitehead tower of $O(3)$

[edit: added analogous discussion to *octahedral group* and *icosahedral group* ]