I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A\_n generate A\_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is that if two even permutations fail to generate A\_n, then they must both be contained in some maximal subgroup of A\_n.


I can't see why this is the case? 


Any help much appreciated.



Dixon's Theorem
If you have two elements of ${\rm A}\_n$ which do not lie both in any maximal subgroup of
${\rm A}\_n$, then they in particular do not lie both in *any* proper subgroup of ${\rm A}\_n$.
This in turn means that the only subgroup of ${\rm A}\_n$ which contains your elements is
${\rm A}\_n$ itself. Hence your two elements *generate* ${\rm A}\_n$.


