Permutations: the product of $(1,2)(1,2,3) = (1,3)$

Question

I'm trying understand why the following permutation is the case. I understand we go from right to left, although others may go from left to right, but I can't seem to produce the correct answer. I keep getting $(1,2,3)$ which is obviously wrong.

Answer 1

We start with $(1$, and build our cycle from that. $1$ gets sent by the left cycle to $2$, and then the right cycle sends that $2$ to $3$. So we write $(1,3$.

Now we look at where $3$ goes. The first cycle doesn't touch it, and then the second cycle sends it to $1$, meaning the cycle is complete with $(1,3)$. Technically we should check $2$ as well, but there is nowhere else to send it, so we are finished.

Answer 2

Product of permutation is composition of maps (right to left). $$1\mapsto2\mapsto1,\:2\mapsto3\mapsto3,\:3\mapsto1\mapsto2$$ $$\begin{pmatrix}1 & 2 & 3\\2 & 1 & 3\end{pmatrix}\begin{pmatrix}1 & 2 & 3\\2 & 3 & 1\end{pmatrix}=\begin{pmatrix}1 & 2 & 3\\1 & 3 & 2\end{pmatrix}$$

Answer 3

Let’s compute the value of the permutation at 1,2 and 3. By definition, $(1,2,3)$ evaluated at $1$ is $2$. $(1,2)$ evaluated at $2$ is $1$. Thus, $(1,2)(1,2,3)$ evaluated at $1$ is $1$. By definition, $(1,2,3)$ evaluated at $2$ is $3$. $(1,2)$ evaluated at $3$ is $3$. So $(1,2)(1,2,3)$ evaluated at $2$ is $3$. By definition, $(1,2,3)$ evaluated at $3$ is $1$. $(1,2)$ evaluated at $1$ is $2$. So $(1,2)(1,2,3)$ evaluated at $3$ is $2$. So we can check that $(1,2)(1,2,3)=(2,3)$.