This is from exercise 19 in chapter 15 of Fraleigh's book "A first course in abstract algebra".

True or false: All nontrivial finite simple groups have prime order

The answer to this is false but I cannot figure out how to prove it.

We can show that if a group $G$ has prime order, then there can be no nontrivial, improper subgroup $H$ as the order of $H$ must divide the order of $G$. I have read on other websites that we can show that a finite simple group does not necessarily have to have prime order by considering the group $A_5$ with order $|A_5|=60$ as a counterexample. I see how this proves the falsity of that statement but I do not understand how one gets the idea to consider this as a counterexample. How would one go about doing this?

The same section contains a proof on the falsity of the converse of the Theorem of Lagrange, i.e. that the existence of a divisor $n$ of the order $|G|$ does not necessarily imply the existence of a subgroup with order $|G|/n$. Is this a key fact to use?

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    $\begingroup$ Lagrange's converse is only relevant in the sense that, if the converse were true, all nontrivial groups of non-prime order would have a proper normal subgroup. But it's really not relevant, I think you're just expected to be somewhat familiar with simple groups, enough to know that nonabelian ones exist, at least. $\endgroup$
    – pjs36
    Apr 8, 2016 at 1:48
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    $\begingroup$ It seems like a strange question to ask at a point in the book where you only have Lagrange's theorem theorem available. Unless Fraleigh does things in a weird order and has already introduced $A_5$ or some other nonabelian finite simple group, I don't see how the reader can be expected to know of or stumble upon its existence. Even defining $A_5$ takes a bit of work. $\endgroup$
    – user169852
    Apr 8, 2016 at 18:04
  • $\begingroup$ After reading your comment I went back and reviewed the chapter. He does indeed offer a theorem in that section that states that the alternating group $A_n$ is simple for $n>4$. I suppose if I had paid attention upon reading it the first time it would have been obvious that this is a finite simple group of composite order. $\endgroup$ Apr 8, 2016 at 18:28


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