Permutations are bijections, so by Theorem 2.9.1 they have inverse functions. The inverse function to a permutation undoes what did, in the sense that if then . In two row notation you write beneath , so you can get the two row notation for by swapping the rows (and reordering).
We know by Theorem 2.9.2 that the composition of two bijections is a bijection, so the composition of two permutations of a set is again a permutation of .
Let
Then is the function whose rule is “do , then do .” Thus
In two row notation,
There are several similarities between composing permutations and multiplying nonzero numbers. For example, if , , and are nonzero real number then . Furthermore the identity permutation behaves for composition just like the number 1 behaves for multiplication. For each nonzero real number we have , and for each permutation we have . Equally, for each nonzero real number there is another nonzero real number such that , and for each permutation there is an inverse permutation such that . Because of these similarities we often talk about multiplying two permutations when we mean composing them, and given two permutations and we usually write for their composition instead of .
Composition has one big difference with real number multiplication: the order matters.
With and as before,
Comparing this to the example in the previous section, and are different. Composition of permutations is not commutative in general.
Two permutations and are said to commute if .