Non-Abelian Simple Groups

Swlabr · #1 $Old$ November 2nd, 2009, 08:11 AM

I was wondering - $A_5$ is the only non-abelian simple group of order less than 100. What is the next one along? I know there is one of order somewhere in the region of 168 (the Projective Special Linear Group $PSL_2(7)$ ), but is there any of order between these two?

tonio · #2 $Old$ November 2nd, 2009, 08:42 AM

Quote:

Originally Posted by Swlabr $View Post$

I was wondering - $A_5$ is the only non-abelian simple group of order less than 100. What is the next one along? I know there is one of order somewhere in the region of 168 (the Projective Special Linear Group $PSL_2(7)$ ), but is there any of order between these two?

\

No, there isn't...non-abelian, that is.

Tonio

Swlabr · #3 $Old$ November 2nd, 2009, 09:05 AM

Quote:

Originally Posted by tonio $View Post$

\

No, there isn't...non-abelian, that is.

Tonio

Okay, but why? The problem cases that I can see are groups of order $p^2qr$ and $p^3qr$ (120 is the only number of this form in our range though, and 168 is of this form). As $60=2^2.3.5$ , we cannot slap a general condition on groups of these orders not being non-simple, as we can with the other orders.

So, I have become a tad stumped as to why this is...

NonCommAlg · #4 $Old$ November 3rd, 2009, 09:31 PM

a useful and easy-to-prove fact is that if $2 < |G| \equiv 2 \mod 4,$ then $G$ is not simple. see Tough Problem from Rotman's Group Theory, 3rd Ed for a proof of a more general result.