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 $X$. This group may be denoted $S_X$, $\Sigma_X$, or $X!$. When $X$ is the finite set $[n]$ with $n$ elements, then its symmetric group is a finite group, and 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 classifying space $B \Sigma(n)$ of the symmetric group on $n$ elements may be presented by $Emb(\{1,\cdots, n\}, \mathbb{R}^\infty)/\Sigma(n)$, the Fadell's configuration space on $n$ unordered points in $\mathbb{R}^\infty$.
Write $\tau_n$ for the rank $n$ vector bundle over this which exhibits the canonical action of $\Sigma(n)$ on $\mathbb{R}^n$, by permutation of coordinates.
The Thom space $B \Sigma(n)^{\tau_n}$ of this bundle appears as the cofficients of the spectral symmetric algebra of the “absolute spectral superpoint” $Sym_{\mathbb{S}} \Sigma \mathbb{S}$ (see Rezk 10, slide 4).
See also (Hopkins-Mahowald-Sadofsky 94, around def. 2.8)
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 and at Platonic 2-group.)
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.
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)
Michael Hopkins, Mark Mahowald, Hal Sadofsky, Constructions of elements in Picard groups, Contemporary mathematics, Volume 158, 1994 in Eric Friedlander, Mark Mahowald (eds.), Topology and Representation theory (doi:10.1090/conm/158/01454)