TITLE: Simplicity of alternating group $A_n$
QUESTION [22 upvotes]: 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...

REPLY [6 votes]: I prefer the proof going via 3-cycles. It is probably the least elegant proof, but it is the one which you probably would have found when considering the problem without any prior knowledge. Also the general strategy generalizes to classical groups as well as Lie groups: Look for elements $g_1$, which cannot be contained in a normal subgroup, then look for elements $g_2$, such that the normal subgroup generated by $g_2$ contains $g_1$, and continue until you run out of elements to check. The disadvantage is that for each new class of groups you have to come up with the right elements, e.g. 3-cycles for $A_n$, transvections for $Sl_n$, but if you choose the element to start with "close to the identity", you will succeed quite often.