Permutation

Definition
A permutation on a set is defined as a bijective function from the set to itself.

The set of all permutations on a set forms a group with the group operation being function composition, the identity element being the identity function, and the inverse of a permutation being the inverse function. This group is termed the symmetric group on the set. A permutation on a set can thus also be defined as an element of the symmetric group on the set.

Two-line notation for permutations
Suppose $$S$$ is a finite set and $$\sigma$$ is a permutation of $$S$$. A two-line notation for $$\sigma$$ is a two-line description that uniquely determines $$\sigma$$. The first row lists the elements of $$S$$. In the second row, we list, below each element of $$S$$, its image under $$\sigma$$.

For instance, consider the permutation on $$\{1,2,3,4,5 \}$$ that sends $$x$$ to $$6 - x$$. The two-line notation for this permutation is:

$$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 3 & 2 & 1 \\\end{pmatrix}$$.

One-line notation for permutations
The one-line notation for a permutation is an abbreviated form of the two-line notation where we write only the second line, because the first line is understood. This happens, for instance, when the elements of $$S$$ come with a standard ordering and it is understood that the first line, if written, would have the elements in that order.

Cycle decomposition for permutations
The cycle decomposition of a permutation is an expression of the permutation as a product of disjoint cycles. For instance, the permutation on $$\{ 1,2,3,4,5 \}$$ has three cycles: $$(1,5)$$, $$(2,4)$$ and $$(3)$$. The cycle decomposition is this $$(1,5)(2,4)(3)$$.