I am teaching an introductory group theory course, and it has come to the inevitable proof that $A\_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "standard" one with $3$-cycles, and the others using primitivity or conjugacy class size estimation. Does anyone have a list of proofs of simplicity of $A\_n$ with comments? I am sure there are a number of different ideas which could be used... In particular, any method that generalizes to some of the other finite simple groups would be of particular interest...



Simplicity of alternating group $A_n$
Just turning comments into an answer.


1. **Iwasawa's Criterion**. This will do $A\_5$ (using the natural action), $A\_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some notes that might be useful are [here](https://nickpgill.github.io/2014/03/11/topics-in-group-theory/).
2. **A list** of the type you asked for [does exist](https://www.math.uconn.edu/~kconrad/blurbs/grouptheory/Ansimple.pdf) thanks to Keith Conrad (it contains five proofs of the simplicity of $A\_n$, for $n\geq 5$.)
3. See the comments for other suggested proofs.


