I am trying to understand if there is a way we can define the $S_n$ group, just like we do for $D_n$. We know that $D_n$ is defined as $D_n =\langle r,s\mid s^2 = e, r^n = e, srs = r^{-1}\rangle$. Can we do a similar definition for $S_n$?
I know that a transposition $(a,b)$ and a $n$-cycle $(1,2,3...,n)$ generate $S_n$ if and only if $\gcd(b-a, n) = 1$. Using this, can we write something on the lines of $S_n =\langle (a,b),r\mid|(a,b)| = 2, |r| = n,\gcd(b-a,n) = 1\rangle$?
I suppose we could add a relation among the two elements mentioned. For example , for $S_4$, let $(a,b) = (12)$ and $r = (1234)$, then the relation $r(a,b)r(a,b)r = (a,b)$ holds. So, can I write $S_4 = \langle(a,b),r\mid|(a,b)| = 2, |r| = n,\gcd(b-a,n) = 1, r(a,b)r(a,b)r = (a,b)\rangle$? Thank you for your help.