## 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 in 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. Figure 2.1.3. Symmetries of a regular $$n$$-gon

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 $$U(\Quat)$$