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added to icosahedral group discussion of the distinction of definitions as one moves up the Whitehead tower of $O(3)$
$\array{ \mathcal{I} &\hookrightarrow& String_{SU(2)} \\ \downarrow && \downarrow \\ 2 I &\hookrightarrow & Spin(3) = SU(2) \\ \downarrow && \downarrow \\ I \simeq A_5 &\hookrightarrow& SO(3) \\ \downarrow && \downarrow \\ I_h \simeq A_5\times \mathbb{Z}/2 &\hookrightarrow & O(3) }$[edit: added analogous discussion to octahedral group and icosahedral group ]
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…
I added a few more points, including the exceptional isomorphisms to Lie-type groups over finite fields.
Thanks for this! I have added the pointers on discrete 2-group enumeration to the References at 2-group
You use ambiguous wording “symmetry” at icosahedral group where you apparently mean “isometry”. Is there a reason ? Should we change to isometry ?
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!
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.
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.
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.
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.)
I added a stubby definition section.
Thanks!
I have copied this also to Platonic Solids – Symmetry groups.
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