The standard proof for the simplicity of the alternating groups consists of three claims. If $n \ge 5$ then (1) all elements of $A_n$ are products of 3-cycles, (2) all 3-cycles in $A_n$ are conjugate, and (3) any nontrivial normal subgroup must contain a 3-cycle. Thus any nontrivial normal subgroup of $A_n$ is equal to $A_n$.
$Question.$ Are there other families of simple groups that would fit a similar argument, replacing "is a 3-cycle" with some property $P$?
In preparation for teaching a group theory class this spring, I'm curious if there is $another$ family (projective special linear groups, for example?) where one could create a proof of simplicity by finding a property $P$ such that (1) the groups in the family are generated by elements with property $P$; (2) elements of property $P$ are conjugate and (3) nontrivial normal subgroups must contain an element of property $P$.