Cyclic notation and product of transpositions

Question

$p=(1234)$ means that $p$ sends $1$ to $2$, $2$ to $3$, $3$ to $4$ and $4$ to $1$. But I dont understand why $p$ is also equal to $(12)(13)(14)$. Isn't that imply $p(1)=2=3=4$?

Answer 1

Read the product of transpositions $(12)(13)(14)$ from right to left $$1234\xrightarrow{(1,4)}4231\xrightarrow{(1,3)}3241\xrightarrow{(1,2)}2341$$ that is $$\begin{pmatrix}1&2&3&4 \\ 2&3&4&1\end{pmatrix}$$ which means $1$ to $2$, $2$ to $3$, $3$ to $4$ and $4$ to $1$.