A permutation is an automorphism in Set. More explicitly, a permutation of a set $X$ is an invertible function from $X$ to itself.
The group of permutations of $X$ (that is the automorphism group of $X$ in $Set$) is the symmetric group (or permutation group) on $X$ (a finite group). This group may be denoted $S_X$, $\Sigma_X$, or $X!$. When $X$ is the finite set $[n]$ with $n$ elements, one typically writes $S_n$ or $\Sigma_n$; note that this group has $n!$ elements.
In combinatorics, one often wants a slight generalisation. Given a natural number $r$, an $r$-permutation of $X$ is an injective function from $[r]$ to $X$, that is a list of $r$ distinct elements of $X$. Note that the number of $r$-permutations of $[n]$ is counted by the falling factorial $n(n-1)\dots(n-r+1)$. Then an $n$-permutation of $[n]$ is the same as a permutation of $[n]$, and the total number of such permutations is of course counted by the ordinary factorial $n!$. (That an injective function from $X$ to itself must be invertible characterises $X$ as a Dedekind-finite set.)
Every permutation $\pi : X \to X$ generates a cyclic subgroup $\langle \pi \rangle$ of the symmetric group $S_X$, and hence inherits a group action on $X$. The orbits of this action partition the set $X$ into a disjoint union of cycles, called the cyclic decomposition of the permutation $\pi$.
For example, let $\pi$ be the permutation on $[6] = \{0,\dots,5\}$ defined by
The domain of the permutation is partitioned into three $\langle\pi\rangle$-orbits
corresponding to the three cycles
Finally, we can express this more compactly by writing $\pi$ in cycle form, as the product $\pi = (0)(1\,3\,5)(2\,4)$.
One may regard the symmetric group $S_n$ as the general linear group in dimension $n$ on the field with one element $GL(n,\mathbb{F}_1)$.
The symmetric groups and alternating groups are the first stages in a restriction of the Whitehead tower of the orthogonal group to “finite discrete ∞-groups” in the sense of homotopy type with finite homotopy groups. The homotopy fibers of the stages of the “finite Whitehead tower” are the stable homotopy groups of spheres (Epa-Ganter 16). (See also at super algebra – Abstract idea.)
on the right: the delooped smooth ∞-group Whitehead tower of the orthogonal group (fivebrane 6-group $\to$ string 2-group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group);
in the middle, its restriction to deloopings of finite groups and their universal ∞-group extensions ($\cdots \to$ covering of alternating group $\to$ alternating group $\to$ symmetric group)
on the left the homotopy fibers of each stage.
Notice that the squares on the right are not homotopy pullback squares. (The homotopy pullback of the string 2-group along $\tilde A \hookrightarrow Spin(n)$ is a $\mathbf{B}U(1)$-extension of $\tilde A$, but here we get the universal finite 2-group extension, by $\mathbb{Z}/24$ instead.