I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to give a simple, convincing example where it's useful to use the two-generating set $\{(1,2),(1,2,...,n)\}$. (I always find it annoying when we teach something and we're not able to convince the students that it's useful.) I asked a couple of colleagues and no simple answer came out -- let me stress that I'd like to find something simple enough, like a remark I could do in passing or an exercise that I could leave to the reader without cheating him/her. Do you know such examples ?



What is the standard 2-generating set of the symmetric group good for?
Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots.


You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a subgroup of $S\_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem from group theory to show that the Galois group contains a $p$-cycle.


Then $f$ has Galois group $S\_p$. This uses the slightly stronger fact that $S\_p$ is generated by any transposition and $p$-cycle (which can be proved from the standard two-generating set).


In turn, constructing a polynomial with Galois group $S\_5$ is useful for proving insolvability of the quintic.


