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There are some similiar questions, but it seems that there is some different equal definition to $A_5$.
The definition I'm using for $A_5$ is that this is the kernel of the sign homomorphism- i.e all the permutations with even number of transpositions.
I would like to get help with the equivalence of this definition , to the definition of subgroup of $S_5$ , generated by all cycles of order 3.

EDIT- THERE IS AN ANSWER ON THE LINK IN COMMENTS

Ron Abramovich
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1 Answers1

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A5 is the group [3,3,3]+ or [3,5]+ in Coxeter notation.

You could make it, I suppose, out of (AB)³ = (BC)³ = (CD)³ = I, and (AC)² = (BD)² = (DA)² = I, which is what the first symbol means.

The second one means (AB)³ = (BC)^5 = (AC)² = I, which is what the second definition.

A5 is a relatively primitive group, but you can indeed generate A5 from purely cycles of three.

wendy.krieger
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  • Can you please descrive the idea in the usual notation of symetrices? – Ron Abramovich Jan 06 '21 at 09:57
  • I don't know what you mean by "usual notation of symmetries". I dabble in higher dimension stuff, not maths. So these are copies from Conway and Coxeter. It's a matter of walking the Cayley diagrams, the omnitrunc pentachoron, and the rhombotruncated icosadodecahedron resp. – wendy.krieger Jan 06 '21 at 10:07