It's well known that every subgroup $G$ of $S\_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)


I have heard many times a stronger result, which is that for all $n>3$, every subgroup $G$ of $S\_n$ has a generating set of size at most $n/2$. Note that this would be tight: for example, $n/2$ disjoint transpositions cannot be minimized further.


However, as far as I can tell, none of the places I have seen this theorem give a proof, nor do their references. Can anyone point me to a proper proof, or give one, for this seemingly important theorem?



Subgroup ranks of the symmetric group
After some googling, one finds a few references. Most point to 


Cameron, Peter J.; Solomon, Ron; Turull, Alexandre, Chains of subgroups in symmetric groups. J. Algebra 127 (1989), no. 2, 340–352. 


Which itself attributes this to Peter Neumann, private communication.


They also say: "As Peter is unlikely to publish his result, we shall sketch a recipe for a proof of Neumann’s theorem here.
" (And then proceed to give a sketch.)


Note that the proof seems to depend on the Classification of Finite Simple Groups (to deal with the case of primitive permutation groups). 


