Product of Permutations Can someone explain how to multiply two permutations? I cannot get myself to understand. Please be very simple and explain each step. 
 A: If you have two permutations displayed in this form:
$$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 2 & 5 & 1\end{pmatrix}$$
and
$$\tau = \begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1\end{pmatrix}$$
you can reorder the columns of $\tau$ so that the first line of $\tau$ matches the second line of $\sigma$, and you obtain $\tau = \begin{pmatrix}3 & 4 & 2 & 5 & 1\\ 3 & 2 & 4 & 1 & 5 \end{pmatrix}$, and $\tau\sigma$ now is described by the first line of $\sigma$ and the last line of $\tau$, i.e 
$$\tau\sigma = \begin{pmatrix}1 & 2 & 3 & 4 & 5\\3 & 2 & 4 & 1 & 5\end{pmatrix}$$
A: Permutations are nothing but functions, and their product is nothing but function composition. In more details, a permutation on the set $S=\{1,2,3,\cdots ,n \}$ is, by definition, a function $\sigma:S\to S$ which is a bijection. The product of two such permutations $\sigma $ and $\tau $ is the function composition $\sigma \circ \tau $. Since the composition of bijective functions is a bijection, it follows that $\sigma \circ \tau $ is indeed again a permutation. 
A common way to represent permutations is using cycle notation. If you are having difficulties with that I suggest you try youtube to find plenty of videos explaining how it works and compute with examples. 
