Alternative proofs that A5A_5 is simple

What different ways are there to prove that the group A_5 is simple?

I’ve collected these so far:

Answer

I'm very happy to show this proof, which makes use of a technique frequently employed in a recent paper of mine. It's my favorite proof that A_5 is not solvable, which as you pointed out in your last bullet proves that A_5 is simple.

Definition. For n\in \mathbb{N}, denote by \pi(n) the set of prime divisors of n. The prime graph of a finite group G, denoted \Gamma_G, is the graph with vertex set \pi(|G|) with an edge between primes p and q if and only if there is an element of order pq in G.

By Lucido (1999), Prop. 1, the complement of the prime graph of a solvable group G is triangle-free. It is obvious from cycle types that A_5 contains no elements of order 6,10, or 15, so \Gamma_{A_5} is the empty graph on three vertices. Therefore, A_5 is not solvable.

Attribution
Source : Link , Question Author : Community , Answer Author : Alexander Gruber

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