Number of Involutions generating Symmetric Group

Question

How to show that strictly less than $n-1$ involutions (of which the transpositions are a special case) could not generate $S_{n}$ for $n > 3$?

I know that $n-1$ transpositions are sufficient, or that $S_n$ could be generated by the two elements $(1 ~ 2), (1 ~ 2 ~ \ldots ~ n)$, but what about less than $n-1$ involutions?

Answer 1

This is not true. For example, $S_{10}$ can be generated by the three involutions (2,3), (3,5)(4,7)(6,9)(8,10), (1,2)(3,4)(5,6)(7,8).

Of course for transpositions the answer is easy, as you need $n-1$ transpositions to be transitive on $n$ points.

Answer 2

It is not true. $S_3$ can be generated by two involution for example.

$<\{(1,2),(2,3)\}> =S_3$.