Let \(n\) be a prime and let \(G \leq S_n\) contain an \(n\)-cycle.
If \(G\) contains a transposition, then \(G = S_n\).
If \(n \geq 3\) and \(G\) contains a \(3\)-cycle, then \(G = A_n\) or \(G = S_n\).
For composite \(n\), a transposition and an \(n\)-cycle need not generate \(S_n\). In \(S_4\), the elements \((1\,3)\) and \((1\,2\,3\,4)\) preserve the partition \(\theset{\theset{1,3},\theset{2,4}}\), and they generate the dihedral group of order \(8\). The specific transposition \((1\,2)\) and the specific \(n\)-cycle \((1\,2\,\cdots\,n)\) do generate \(S_n\) for every \(n \geq 2\).
For composite \(n\), a \(3\)-cycle and an \(n\)-cycle need not generate a group containing \(A_n\). In \(S_6\), both \((1\,2\,3)\) and \((1\,4\,2\,5\,3\,6)\) preserve the partition \(\theset{\theset{1,2,3},\theset{4,5,6}}\), so the group they generate lies in the stabilizer of that partition, which has order \(72 < \abs{A_6} = 360\).
For the Galois group \(G \leq S_n\) of an irreducible separable polynomial \(f\) of degree \(n\), the action on the roots is transitive, so \(n \divides \abs{G}\) by the orbit-stabilizer theorem. If \(n\) is prime, Cauchy’s theorem gives an element of order \(n\) in \(G\), and every element of order \(n\) in \(S_n\) is an \(n\)-cycle.