# Finite Simple Groups other than $A_n$ and $\rm{PSL}_n$

The finite simple groups taught in undergraduate or graduate courses are only up to $A_n$ or $\rm{PSL}_n$. Even many undergraduate and graduate texts do not consider simple groups beyond these two (one such graduate text I saw is Groups and Representations-Alperin, Bell).

What could be the other families of simple groups which can be introduced in undergraduate, assuming that the students know only Group Theory (up to Sylow theorems) and possibly Linear Algebra? And what is better, simple way of introducing it for undergraduates?

• Cyclic finite groups of prime order, perhaps the Mathieu groups. – Pedro Tamaroff Jan 23 '16 at 5:02
• there are various ways for introducing Mathieu groups but I didn't see which way is easiest. – p Groups Jan 23 '16 at 5:04
• "Easiest" is subjective: what might be intuitive and reasonable for some may very well be terse for others. – Pedro Tamaroff Jan 23 '16 at 5:18
• It's not particularly difficult to define other classical groups, like unitary groups and symplectic groups, but this is probably not a good idea in an undergraduate course, because the details are similar to those for the linear groups, but more technically complicated. I would go for the Mathieu groups. – Derek Holt Jan 23 '16 at 10:08