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# Section6.1Introduction to permutation groups¶ permalink

##### Definition6.1.1

A permutation on a set $A$ is a bijection from $A$ to $A\text{.}$ We say a permutation $\sigma$ on $A$ fixes $a\in A$ if $\sigma(a)=a\text{.}$

##### Example6.1.2

Let $A$ be the set $A=\{\Delta, \star, 4\}\text{.}$ Then the functions $\sigma : A\to A$ defined by

\begin{equation*} \sigma(\Delta)=\star, \sigma(\star)=\Delta, \text{and } \sigma(4)=4; \end{equation*}

and $\tau : A\to A$ defined by

\begin{equation*} \tau(\Delta)=4, \tau(\star)=\Delta, \text{and } \tau(4)=\star \end{equation*}

are both permutations on $A\text{.}$

##### Definition6.1.3

Composition of permutations on a set $A$ is often called permutation multiplication, and if $\sigma$ and $\tau$ are permutations on a set $A\text{,}$ we usually omit the composition symbol and write $\sigma \circ \tau$ simply as $\sigma \tau\text{.}$

##### Warning6.1.4

For us, if $\sigma$ and $\tau$ are permutations on a set $A\text{,}$ then applying $\sigma \tau$ to $A$ means first applying $\tau$ and then applying $\sigma\text{.}$ This is due to the conventional right-to-left reading of function compositions.

That is, if $a\in A\text{,}$ by $\sigma \tau(a)$ we mean $\sigma(\tau(a))\text{.}$ (Some other books/mathematicians do not use this convention, and read permutation multiplication from left-to-right. Make sure to always know what convention your particular author or colleague is using!)

##### Example6.1.5

Let $A\text{,}$ $\sigma\text{,}$ and $\tau$ be as in Example 6.1.2. Then $\sigma \tau$ is the function from $A$ to $A$ defined by

\begin{equation*} \sigma \tau(\Delta)=4, \sigma \tau(\star)=\star, \text{and } \sigma \tau(4)=\Delta, \end{equation*}

while $\tau \sigma$ is the function from $A$ to $A$ defined by

\begin{equation*} \tau \sigma (\Delta)=\Delta, \tau \sigma (\star)=4, \text{and } \tau \sigma(4)=\star \end{equation*}
##### Definition6.1.6

Given a set $A\text{,}$ we define $S_A$ to be the set of all permutations on $A\text{.}$

##### Proof

We will in the future use language provided by the following definition:

##### Definition6.1.8

A group is said to be a permutation group if it is a subgroup of $S_A$ for some set $A\text{.}$

##### Remark6.1.9

Notice that if $A$ and $B$ are sets, then $|A|=|B|$ if and only if $S_A\simeq S_B\text{.}$

Thus, for any set $B$ with $|B|=n \in \Z^+\text{,}$ we have $S_B\simeq S_A\text{,}$ where $A=\{1,2,\ldots,n\}\text{.}$ Since we are concerned in this course primarily with group structures which are invariant under isomorphism, we may focus now on groups of permutations on the set $\{1,2,\ldots, n\}$ ($n\in \Z^+$).