Can $S_4$ be generated by $(1,2),(1,3),(1,4)$? I am having troubles answering it. It sometimes confuses me. I could really use some help.
 A: For example, do you know that in general $\;S_n=\langle\;(1\;2)\;,\;\;(1\;2\;\ldots n)\;\rangle\;$ ? If so, observe that
$$(1 4)(13)(12)=(1234)$$
and thus...
A: Another approach/answer is to check that with those three transpositions one can make the rest of them in $S_{4}. After that check that with all the transpositions you can make all the permutations. 
A: Hint:
Say you have three persons $A$, $B$ and $C$. Initially they are seated on seats $1$, $2$ and $3$. Now, say you want to have persons $B$ and $C$ exchange seats, that is, $B$ should get to seat $3$ and $C$ to seat $2$. However, the only moves allowed is to have the persons on seats $1$ and $2$ exchange seats, or the persons on seats $1$ and $3$ exchange seats. How to switch $B$ and $C$ by performing several of the allowed moves?
In general, you can permute them in any way, with seat $1$ as "processing center".
A: Assuming you already know that $S_4$ is generated by all transpositions like $(ij)$ (here $i,j \in \{1,2,3,4\}$ and $i<j$), it is sufficient to show that $(ij)$ can be represented as a product of $(12),(13),(14)$. Then consider the following equality:
$$
  (ij)=(1j)\cdot(1i)\cdot(1j).
$$
Update: Any permutation can be presented as a product of independent cycles. And any cycle can be presented as a product of transpositions $(ij)$. For example, 
$$
  (a_1 a_2 \ldots a_k) = (a_1 a_2)(a_1 a_3) \ldots (a_1 a_k).
$$
