The alternating group on five elements is simple.
Recall that has order , so Lagrange's theorem states that any subgroup of has order dividing .
Suppose is a normal subgroup of , which is not the trivial subgroup . If has order divisible by , then by Cauchy's theorem there is a -cycle in (because the -cycles are the only elements with order in ). Because is a union of conjugacy classes, and because the -cycles form a conjugacy class in for , would therefore contain every -cycle; but then it would be the entire alternating group.
If instead has order divisible by , then there is a double transposition such as in , since these are the only elements of order in . But then contains the entire conjugacy class so it contains every double transposition; in particular, it contains and , so it contains . Hence as before contains every -cycle so is the entire alternating group.
So must have order exactly , by Lagrange's theorem; so it contains an element of order since prime order groups are cyclic.
The only such elements of are -cycles; but the conjugacy class of a -cycle is of size , which is too big to fit in which has size .