group theory

# Contents

## Idea

A permutation is an automorphism in Set. More explicitly, a permutation of a set $X$ is an invertible function from $X$ to itself.

## Definition

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.)

## Concrete representations

### Via cycle decompositions

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

$\pi = \array{0 \mapsto 0 \\ 1 \mapsto 3 \\ 2 \mapsto 4 \\ 3 \mapsto 5 \\ 4 \mapsto 2 \\ 5 \mapsto 1}$

The domain of the permutation is partitioned into three $\langle\pi\rangle$-orbits

$[6] = \{0\} \cup \{1,3,5\} \cup \{2,4\}$

corresponding to the three cycles

$0 \underset{\pi}{\to} 0 \qquad 1 \underset{\pi}{\to} 3 \underset{\pi}{\to} 5 \underset{\pi}{\to} 1 \qquad 2 \underset{\pi}{\to} 4 \underset{\pi}{\to} 2$

Finally, we can express this more compactly by writing $\pi$ in cycle form, as the product $\pi = (0)(1\,3\,5)(2\,4)$.

## Properties

### Relation to the field with one element

One may regard the symmetric group $S_n$ as the general linear group in dimension $n$ on the field with one element $GL(n,\mathbb{F}_1)$.

### Whitehead tower and relation to supersymmetry

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.)

$\array{ && && \vdots \\ && && \downarrow \\ && && \mathbf{B}Fivebrane(n) \\ && \downarrow && \downarrow \\ \mathbf{B}^2 \pi_3 \mathbb{S} &\stackrel{}{\longrightarrow} & \mathbf{B}\mathcal{A}_n &\hookrightarrow& \mathbf{B} String(n) \\ && \downarrow && \downarrow \\ \mathbf{B} \pi_2 \mathbb{S} &\longrightarrow & \mathbf{B} \tilde A_n &\hookrightarrow& \mathbf{B} Spin(n) \\ && \downarrow && \downarrow \\ \pi_1 \mathbb{S} &\longrightarrow& \mathbf{B} A_n &\hookrightarrow& \mathbf{B} SO(n) \\ && \downarrow && \downarrow \\ && \mathbf{B} S_n &\hookrightarrow& \mathbf{B} O(n) }$

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.

## References

Revised on January 21, 2016 16:12:24 by Urs Schreiber (195.37.209.180)