# Group Action of Symmetric Group

## Theorem

Let $\N_n$ denote the set $\set {1, 2, \ldots, n}$.

Let $\struct {S_n, \circ}$ denote the symmetric group on $\N_n$.

The mapping $*: S_n \times \N_n \to \N_n$ defined as:

$\forall \pi \in S_n, \forall n \in \N_n: \pi * n = \map \pi n$

is a group action.

### Group Action on Subset of $\N_n$

Let $r \in \N: 0 < r \le n$.

Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$:

$B_r := \set {S \subseteq \N_n: \card S = r}$

Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as:

$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$

where $\pi \sqbrk S$ denotes the image of $S$ under $\pi$.

Then $*$ is a group action.

## Proof

The group action axioms are investigated in turn.

Let $\pi, \rho \in S_n$ and $n \in \N_n$.

Thus:

 $\ds \pi * \paren {\rho * n}$ $=$ $\ds \pi * \map \rho n$ Definition of $*$ $\ds$ $=$ $\ds \map \pi {\map \rho n}$ Definition of $*$ $\ds$ $=$ $\ds \map {\paren {\pi \circ \rho} } n$ Definition of Composition of Mappings $\ds$ $=$ $\ds \paren {\pi \circ \rho} * n$ Definition of $*$

demonstrating that Group Action Axiom $\text {GA} 1$ holds.

Then:

 $\ds I_{\N_n} * n$ $=$ $\ds \map {I_{\N_n} } n$ where $I_{\N_n}$ is the identity mapping on ${\N_n}$ $\ds$ $=$ $\ds n$ Definition of Identity Mapping

demonstrating that Group Action Axiom $\text {GA} 2$ holds.

$\blacksquare$