Q:

A difficulty in understanding a statement in the method for finding all normal subgroups of the symmetric group $S_{4}$

I was reading this question :
Normal subgroups of $S_4$
And I had a difficulty in understanding this statement:
"Now, any subgroup that contains all transpositions is the whole group."
Could anyone explain for me why this statement is true please?

A:

The transpositions generate the symmetric group: each permutation is a product
of transpositions. For example, a cycle
$$(a_1\ a_2\ \cdots\ a_k)=(a_1\ a_2)(a_2\ a_3)\cdots(a_{k-1}\ a_k).$$

