Generator of factor group $S_4/V_4$

Question

Let $G= S_4$ and $N=V_4=\{1, (12)(34), (13)(24), (14)(23)\}.$ Then $N \lhd G$.

May I know why is $G/N$ generated by $\left\{N(123), N(12)\right\} ?$ I tried to check for $N(34)$ to no avail. Appreciate your advice, thank you.

Answer 1

$$N(3,4)=(N(1,2)(3,4))(3,4)=N(1,2),$$ so $N(3,4)$ is one of the generators.

Answer 2

$|G/N|=24/4=6$. So $G/N$ is either a cyclic group (i.e. isomorphic to $\Bbb{Z}_6$) or it is isomorphic to $S_3$.

If it is not cyclic (check how many elements of order $2$ does it have--more than one???), then like $S_3$ (or $D_3$) it should have two generators (rotation, reflection (think in terms of Dihedral).

Answer 3

$N(123)$ and $N(12)$ can be seen to have orders $3$ and $2$ respectively, since $3$ and $2$ are the smallest powers of $(123)$ and $(12)$ contained in $N$.

Also note that by Lagrange the cyclic groups they generate intersect trivially.

Thus we have $4$ elements right away.

By Lagrange's theorem, $\langle N(123),N(12)\rangle$ has to have order dividing $6$. Since that order is $\ge4$, it equals $6$.