The group of rigid motions of an icosahedron.

Question

Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$.

Can you help me to prove this?

What I have done is shown that the order of the group of rigid motions of icosahedron is 60, which is same as $A_5$.

Show your working so far. It'll both help you and encourage others to help you. — Shaun, Nov 02 '13 at 12:09
what i have found is just the order of the group.. nothing more.. — user104875, Nov 02 '13 at 12:17
what do you know about the group? Do you know that it is simple? — Prahlad Vaidyanathan, Nov 02 '13 at 12:34
yes, A_5 is simple group. so what is the relation of the truth that it is simple and the problem? — user104875, Nov 02 '13 at 12:37
No - not $A_5$. Do you know if the icosahedral group is simple? — Prahlad Vaidyanathan, Nov 02 '13 at 12:39
I don't know the structure of the icosahedral group at all.I know just the order — user104875, Nov 02 '13 at 12:41

Shaun · Accepted Answer · 2014-01-04T11:33:04.313

4

You can do this by considering the dodecahedron, which is dual to the icosahedron and hence has the same group of symmetries.

There are five cubes that fit inside a dodecahedron in such a way that the rotational symmetries permute these cubes with $3$-cycles.

enter image description here

But $3$-cycles in $\mathcal{S}_{5}$ generate $\mathcal{A}_{5}$.

I hope that gives you some strong clues :)

edited Jan 04 '14 at 11:33

answered Nov 02 '13 at 12:37

Shaun

Here's a picture: http://www.chiark.greenend.org.uk/~sgtatham/polypics/dodec-cubes-large.jpeg. – Shaun Nov 02 '13 at 12:42
1

this is very powerful clue! thx a lot – user104875 Nov 02 '13 at 12:45
You're welcome :) – Shaun Dec 17 '13 at 11:23
1

You can also work directly with the icosahedron. First inscribe five octahedra within by this method. Then use rotations of the triangular faces to prrmute these. – Oscar Lanzi Aug 15 '22 at 16:43

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