subgroups with cycles

Question

Let Sym(n) denote the symmetric group on n letters and let H be a subgroup of Sym(n) .
Suppose that H contains a k cycle for each value of k from 2 through n .
This should be enough to conclude that H = Sym(n) .

Is there a direct proof of this (without appealing to a list of maximal subgroups)?

Thanks in advance.

Answer 1

The existence of an $n$-cycle shows that $H$ is transitive. Then the existence of an $(n-1)$-cycle shows that it is $2$-transitive. .... Prove by induction on $r$ that the existence of the $(n-r)$-cycle implies that $H$ is $(r+1)$-transitive.