# Show the falsity of the statement "all nontrivial finite simple groups have prime order"

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?

• 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. Apr 8, 2016 at 1:48
• 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.
– user169852
Apr 8, 2016 at 18:04
• 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. Apr 8, 2016 at 18:28