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 $\pi = (0)(1\,3\,5)(2\,4)$.
Let $n \in \mathbb{N}$ and $\Sigma(n)$ the symmetric group on $n$ elements. Then the conjugacy classes of elements of $\Sigma(n)$, hence of permutations of $n$ elements, correspond to the cycle structures: two elements are conjugate to each other precisely if they have the same number of distinct cycles of the same length, or in other words if they define the same underlying partition of $n$.
For the symmetric group on three elements there are three such classes:
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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)
Last revised on July 20, 2017 at 04:11:58. See the history of this page for a list of all contributions to it.