Permutation Groups on a Set

# Permutation Groups on a Set

Recall from the The Symmetric Groups on n Elements page that if $\{ 1, 2, ..., n \}$ is the $n$-element set of positive integers and if $S_n$ denotes the set of all permutations on $\{ 1, 2, ..., n \}$ then $(S_n, \circ)$ where $\circ$ is the operation of function composition defines a group called the symmetric group on $n$-elements.

Now suppose that $X$ is any set. We can analogously define a group on the set of permutations on $X$.

 Definition: Let $X$ be any set and let $S_X$ denote the set of all permutations on $X$. Then $(S_X, \circ)$ is called the Permutation Group on $X$.

Note that if $X = \{ 1, 2, ..., n \}$ then $S_X = S_n$ is the permutation group on $n$-elements. If $X = A = \{ x_1, x_2, ..., x_n \}$ is a general $n$-element set then $S_X = S_A$ is a symmetric group on a general $n$-element set $A$. Of course, what's more interesting is when $X$ is a countably infinite or uncountably infinite set.

For example, consider the set $X = \mathbb{Z}$. Then $S_{X} = S_{\mathbb{Z}}$ is the set of all permutations on $\mathbb{Z}$. For example, the following functions $\sigma : \mathbb{Z} \to \mathbb{Z}, \delta : \mathbb{Z} \to \mathbb{Z} \in S_{\mathbb{Z}}$ are permutations on $\mathbb{Z}$ as you should verify:

(1)
\begin{align} \quad \sigma (x) = x + 1 \end{align}
(2)
\begin{align} \quad \delta(x) = x - 1 \end{align}

Then the composition $\sigma \circ \delta$ is given for all $x \in \mathbb{Z}$ by

(3)
\begin{align} \quad (\sigma \circ \delta)(x) = \sigma(\delta(x)) = \sigma(x - 1) =(x - 1) + 1 = x \end{align}

Therefore we see that $\sigma \circ \delta = i$ where $i$ is the identity permutation on $\mathbb{Z}$.