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?