Product of Permutations

A Product of Permutations is very similar to function composition. You work right-to-left, passing the set through each permutation to get what it maps to. Eventually, it’ll loop around, and that is a cycle. Then, you go to the next smallest disjoint element and repeat until all cycles are complete.

For example, if we have $\sigma =(123)$ and $\rho =(1534)$ in $S_6$, then we can iterate the elements in $S_6=\{1,2,3,4,5,6\}$ first through $\rho$, and then through $\sigma$, to find all the new mapping cycles:

$$\begin{array}{rlrl}\sigma \rho 1& =(123)(1534)(1)& =(123)(5)& =(5)\\ \to \sigma \rho 5& =(123)(1534)(5)& =(123)(3)& =(1)\\ \to \dots & & & \end{array}$$

And so on, until you complete a cycle, then you continue with the next smallest number. In this example, you would get $(15)(234)(6)$ - try it yourself! Normally, we do this process mentally, too, so it looks a lot like just writing out the answer immediately - just feed the numbers through the cycles.

Products of permutations are not commutative - this is provable the same way as function composition - unless disjoint, as this is like the identity element; unchanging.