## Section2.1Examples of groups

Groups are one of the most basic algebraic objects, yet have structure rich enough to be widely useful in all branches of mathematics and its applications. A group is a set $G$ with a binary operation $G\times G \to G$ that has a short list of specific properties. Before we give the complete definition of a group to the next section (see Definition 2.2.1), this section introduces examples of some important and useful groups.

### Subsection2.1.1Permutations

A permutation of a set $X$ is a bijection from $X$ to itself, that is, a function that is both one-to-one and onto. Given two permutations $\alpha,\beta$ of a set $X\text{,}$ we write $\alpha\beta$ to denote the composition of functions $\alpha\circ\beta\text{.}$

###### Definition2.1.1.

Let $X$ be a set and let $\Perm(X)$ denote the set of all permutations of $X\text{.}$ The group of permutations of $X$ is the set $G=\Perm(X)$ together with the binary operation $G\times G\to G$ given by function composition, that is, $(\alpha,\beta) \to \alpha\circ\beta\text{.}$ For the special case $X = \{1,2,\ldots,n\}$ for some integer $n\geq 1\text{,}$ the group $\Perm(X)$ is called the symmetric group, and is denoted $S_n\text{.}$

A permutation $\sigma$ of $X=\{1,2,\ldots,n\}$ can be specified by a list of values $[\sigma(1),\sigma(2),\ldots,\sigma(n)]\text{.}$ 1  For example, the list $[3,1,2]$ specifies the permutation $\sigma\colon \{1,2,3\}\to\{1,2,3\}$ given by

\begin{equation*} \sigma(1)=3, \; \sigma(2)=1,\; \sigma(3)=2. \end{equation*}

Let $\tau=[2,1,3]\text{.}$ Find $\sigma\tau\text{,}$ $\tau\sigma\text{,}$ and $\sigma^2=\sigma\sigma\text{.}$

There is a reason that we use square brackets, rather than parentheses, for the list of output values of a permutation: Lists written with parentheses are used, by nearly universal convention, for a different representation of permutations called cycle notation. Cycle notation is introduced in Exercise 2.2.2.7.

$\sigma\tau= [1,3,2]\text{,}$ $\tau\sigma=[3,2,1]\text{,}$ $\sigma^2=[2,3,1]$

### Subsection2.1.2Symmetries of regular polygons

Informally and intuitively, we say that regular polygons have rotational and mirror symmetries. Specifically, the rotational symmetries are rotations about the center $O$ of the polygon, clockwise or counterclockwise, by some angle $\angle POP'\text{,}$ where $P,P'$ are any two vertices. The mirror symmetries of the polygon are reflections across lines of the form $\overline{OP}$ or $\overline{OM}\text{,}$ where $P$ is any vertex and $M$ is the midpoint of any edge of the polygon. See Figure 2.1.3.

Here are some standard notations for rotations and reflections in the plane. See Figure 2.1.4.

###### Rotations in the plane.

Fix a center point $O\text{.}$ We write $R_\theta$ to denote the rotation by angle $\theta$ about the point $O\text{.}$ Angle units can be revolutions or degrees or radians, whatever is most convenient. We observe the usual convention that positive values of $\theta$ denote counterclockwise rotations and negative values of $\theta$ denote clockwise rotations.

###### Reflections in the plane.

We write $F_L$ to denote the reflection across the line $L\text{.}$ This means that $P'=F_L(P)$ if and only if $\overline{PP'}\perp L$ and the distance from $P$ to $L$ is the same as the distance from $P'$ to $L\text{.}$

Given symmetries $A,B\text{,}$ we write $AB$ to denote the composition $A\circ B\text{.}$ For example, for the symmetries of the equilateral triangle, with angles in degrees, and with $L=\overline{OP}$ for some vertex $P\text{,}$ we have $R_{240}R_{120}=R_{0}$ and $F_LR_{120}=R_{-120}F_L\text{.}$

###### Definition2.1.5.

The dihedral group, denoted $D_n$ , is the set of rotation and reflection symmetries of the regular $n$-gon together with the binary operation of function composition.

Let $X$ be the square centered at the origin in the $x,y$-plane with vertices at $(\pm 1,\pm 1)\text{.}$ The square $X$ has lines of symmetry $H,V,D,D'$ (horizontal, vertical, diagonal, and another diagonal) where $H,V$ denote the $x,y$ axes, respectively, and $D,D'$ denote the lines $y=-x,y=x\text{,}$ respectively. See Figure 2.1.7.

The symmetries of the square $X$ are

\begin{equation*} D_4=\{R_0,R_{1/4},R_{1/2},R_{3/4},F_H,F_V,F_D,F_{D'}\} \end{equation*}

where the rotation angles units are revolutions. Find the following.

1. $\displaystyle R_{1/4}R_{1/2}$
2. $\displaystyle R_{1/4}F_H$
3. $\displaystyle F_HR_{1/4}$
4. $\displaystyle F_HF_D$
5. $\displaystyle F_DF_H$
6. $\displaystyle (F_DR_{1/2})^2= F_DR_{1/2}F_DR_{1/2}$
7. $\displaystyle (F_DR_{1/2})^3$
1. $\displaystyle R_{3/4}$
2. $\displaystyle F_{D'}$
3. $\displaystyle F_D$
4. $\displaystyle R_{1/4}$
5. $\displaystyle R_{3/4}$
6. $\displaystyle R_0$
7. $\displaystyle F_{D'}$

### Subsection2.1.3The norm 1 complex numbers

###### Definition2.1.8.

The circle group, denoted $S^1\text{,}$ is the set

\begin{equation*} S^1=\{z\in \C\colon |z|=1\} \end{equation*}

of norm 1 complex numbers together with the binary operation $S^1\times S^1\to S^2$ given by complex multiplication, that is, $(z,w)\to zw\text{.}$

Show that if $z,w$ are elements of $S^1\text{,}$ then their product $zw$ is also in $S^1\text{.}$

### Subsection2.1.4The $n$-th roots of unity

Let $n\geq 1$ be an integer. The set

\begin{equation*} C_n=\{z\in \C\colon z^n=1\} \end{equation*}

is called the set of (complex) $n$-th roots of unity.

1. Let $\omega=e^{i2\pi/n}\text{.}$ Show that $\omega^k$ is in $C_n$ for all integers $k\text{.}$
2. Show that, if $z$ is an element of $C_n\text{,}$ then $z=\omega^k$ for some integer $k\text{.}$
3. Show that the set $C_n$ consists of precisely the $n$ elements
\begin{equation*} \{\omega^0,\omega^1,\omega^2,\ldots,\omega^{n-1}\}. \end{equation*}
###### Definition2.1.11.

The set $C_n=\{\omega^0,\omega^1,\omega^2,\ldots,\omega^{n-1}\}$ , together with the operation of complex multiplication, is called the group of $n$-th roots of unity.

### Subsection2.1.5Integers

###### Definition2.1.12.
The set $\Z$ of integers, together with the operation of addition, is called the group of integers. Similarly, the set $\Z_n$ of integers modulo $n$ (where $n$ is some integer $n\geq 1$), together with the operation of addition modulo $n\text{,}$ is called the group of integers mod $n$.

### Subsection2.1.6Invertible matrices

Let $n\geq 1$ be an integer. We write $GL(n,\R)$ to denote the set of $n\times n$ invertible matrices with real entries. We write $GL(n,\C)$ to denote the set of $n\times n$ invertible matrices with complex entries.

###### Definition2.1.13.

The set $GL(n,\R)$ (respectively, $GL(n,\C)$), together with the binary operation of matrix multiplication, is called the group of $n\times n$ real (respectively, complex) invertible matrices, or also the general linear group.

### Subsection2.1.7Nonzero elements in a field

Let $\F$ be a field, such as the rational numbers $\Q\text{,}$ the real numbers $\R\text{,}$ or the complex numbers $\C\text{.}$ We write $\F^\ast$ to denote the set of nonzero elements in $\F\text{.}$

###### Definition2.1.14.
Let $\F$ be a field. The set $\F^\ast$, together with the binary operation of multiplication, is called the group of nonzero elements in the field $\F\text{.}$

### Subsection2.1.8Unit quaternions

###### Definition2.1.15.
The set $U(\Quat)$ of quaternions of norm 1 (defined in Subsection 1.2.4), together with the binary operation of quaternion multiplication, is called the group of unit quaternions.

### Exercises2.1.9Exercises

###### 1.Matrices for the dihedral group $D_4$.
Let $H$ denote the $x$-axis in the $x,y$-plane. The map $F_H\colon \R^2\to \R^2$ is a linear map whose matrix is $\displaystyle \left[\begin{array}{cc}1 \amp 0\\ 0 \amp -1\end{array}\right]\text{.}$ The map $R_{1/4}\colon \R^2\to \R^2$ is a linear map whose matrix is $\displaystyle \left[\begin{array}{cc}0 \amp -1\\ 1 \amp 0\end{array}\right]\text{.}$ Find the matrices for the remaining elements of the dihedral group $D_4$ as specified in Checkpoint 2.1.6.
###### 2.Complex number operations for the dihedral group $D_4$.

Let $H$ denote the real line $\R$ in the complex plane $\C\text{.}$ The map $F_H\colon \C\to \C$ is complex conjugation $z\to {z}^\ast\text{.}$ The map $R_{1/4}\colon \C\to \C$ is the map $z\to e^{i\pi/2}z=iz\text{.}$ Find the maps $\C\to \C$ for the remaining elements of the dihedral group $D_4$ as specified in Checkpoint 2.1.6.

###### 3.

Recall that a binary operation $(x,y) \to x\ast y$ is commutative if $x\ast y=y\ast x$ for all possible values of $x,y\text{.}$

1. Which of the group operations in the examples in this section are commutative? Which are not?
2. Show that $S_n$ is not commutative for $n\gt 2\text{.}$
###### 4.

One of the properties of a group is the existence of an identity element. This is a group element $e$ with the property that $eg=ge=g$ for every $g$ in $G\text{.}$ Find an identity element for each of the groups in the examples in this section.

###### 5.

One of the properties of a group is the existence of an inverse element for every element in the group. This means that for every $g$ in a group $G\text{,}$ there is an element $h$ with the property that $gh=hg=e\text{,}$ where $e$ is the identity element of the group. Find inverses for the following list of group elements.

1. $[4,2,1,3]$ in $S_4$
2. $R_{120}$ in $D_6$ (where $120$ is in degrees)
3. $\frac{1}{\sqrt{2}}(-1+i)$ in $S^1$
4. $7$ in $\Z$
5. $7$ in $\Z_9$
6. $\displaystyle \left[\begin{array}{cc}1 \amp 2\\ 2 \amp 1\end{array}\right]$ in $GL(2,\R)$
7. $r=a+bi+cj+dk$ in $U(\Quat)$