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Prove that there are no nonabelian simple groups of order < 60.

I am studying abstract algebra from Artin and trying some questions and got struck on this.

Let on the contrary there exists a nonabelian simple groups of order <60.

I am at loss of ideas on which result to use and would very much appreciate an outline of solution.

  • It is well-known that the alternating group $A_5$ of order $60$ is the smallest nonabelian simple group. But now that you mention it, I don't know a good way to prove it other than brute force of some sort. – Arthur Sep 02 '21 at 18:11
  • From the answers to the duplicate: "Here's a possibly shorter way to do the problem from "scratch" (assuming only Sylow's theorem, and not assuming Burnside's $pq$-theorem)". – Dietrich Burde Sep 02 '21 at 18:13
  • The question is popular, and there are many different answers. But I agree, it is difficult to sort out a suitable answer. For this you need to say what exactly you are allowed to use. Often just writing " I am at loss of ideas" leads to a "close", see also this post. – Dietrich Burde Sep 02 '21 at 18:16

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