# Section2.6Examples of groups/nongroups, Part II¶ permalink

##### Example2.6.1

Let $n\in \Z^+\text{.}$ We define $n\Z$ by

\begin{equation*} n\Z=\{nx: x\in \Z\}: \end{equation*}

that is, $n\Z$ is the set of all (integer) multiples of $n\text{.}$

##### Remark2.6.3

When we are discussing a group $n\Z\text{,}$ assume that $n\in \Z^+\text{,}$ unless otherwise noted.

We use an example from our next class of groups all the time; in fact, most six-year-olds do as well, since it is used when telling time! Before we get to the example, we need some more definitions and some notation. Throughout the following discussion, assume $n$ is a fixed positive integer.

##### Definition2.6.4

We say integers $a$ and $b$ are congruent modulo [or mod] $n$ if $n$ divides $a-b\text{.}$ If $a$ and $b$ are congruent mod $n\text{,}$ we write $a \equiv_n b\text{.}$

##### Example2.6.5

$1, 7, 13,$ and $-5$ are all congruent mod $6\text{.}$

The following is a profoundly useful theorem; it's so important, it has a special name. We omit the proof of this theorem, but direct interested readers to for, instance, p. 5 in [3].

It follows that for each positive integer $n$ and integer $a\text{,}$ there exists a unique element $R_n(a)$ (the $r$ in the above theorem) of the set $\{0,1,2,\ldots, n-1\}$ such that $a$ is congruent to $R_n(a)$ modulo $n\text{.}$ For example, $R_3(4)=1\text{,}$ $R_3(0)=0\text{,}$ $R_3(17)=2\text{,}$ and $R_3(-5)=1\text{.}$

##### Definition2.6.7

$R_n(a)$ is the remainder when we divide $a$ by $n\text{.}$ (Note: You were probably already familiar with the remainder when you divide a positive integer by $n\text{.}$)

##### Definition2.6.8

We define addition modulo $n$, $+_n\text{,}$ on $\Z$ by, for all $a,b\in \Z\text{,}$

\begin{equation*} a+_n b=R_n(a+b), \end{equation*}

that is, the unique element of $\{0,1,\ldots, n-1\}$ that's congruent to the integer $a+b$ modulo $n\text{.}$

##### Remark2.6.9

Addition mod 24 is what we use to tell time!

The set $\{0,1,2,\ldots, n-1\}$ of remainders when dividing by $n$ is so important we give it a special notation.

##### Definition2.6.10

We define $\Z_n$ to be the set $\{0,1,2,\ldots,n-1\}\text{.}$

##### Warning2.6.11

Note that by our definition of $\Z_n\text{,}$ the integer $n$ itself is not in $\Z_n\text{!}$

We are now ready to consider our next type of group.

##### Example2.6.12

For each $n\in \Z^+\text{,}$ $\langle \Z_n,+_n\rangle$ is a group, called the cyclic group of order $n$ (we will see later why we use the word “cyclic” here). This group is abelian and of order $n\text{.}$

##### Remark2.6.13

In practice, we often omit the subscript $n$ and just write $+$ when discussing addition modulo $n$ on $\Z_n\text{.}$

##### Warning2.6.14

Do not confuse $n\Z$ and $\Z_n\text{!}$ They are very different as sets and as groups.

##### Example2.6.15

In the group $\langle \Z_8,+\rangle$ (where, as indicated by our above remark, $+$ means addition modulo $8$), we have, for instance, $3+7=2$ and $7+7=6\text{.}$ The numbers 2 and 6 are each other's inverse, and $7^{-1}=1\text{.}$ The number $0$ has inverse $0$ (it can't be $8\text{,}$ since $8\not\in \Z_8\text{!}$).

##### Definition2.6.16

For $n\in \Z^+\text{,}$ we define multiplication modulo $n$, denoted $\cdot_n\text{,}$ on $\Z_n$ by $a\cdot_n b = R_n(ab)\text{,}$ the remainder when $ab$ is divided by $n\text{.}$

##### Remark2.6.17

$\Z_n$ is never a group under $\cdot_n$ (do you see why?).

But we can consider the following

##### Definition2.6.18

For $n\in \Z^+\text{,}$ we define $\Z_n^{\times}$ to be the set

\begin{equation*} \{a\in \Z_n\,:\,\gcd(a,n)=1\}\text{.} \end{equation*}
##### Example2.6.19

$\langle \Z_n^{\times},\,\cdot_n\, \rangle$ is a group under multiplication. We omit the proof.

We end this section by considering a few more examples.

##### Example2.6.20

Let $F$ be the set of all functions from $\R$ to $\R\text{,}$ and define pointwise addition $+$ on $F$ by

\begin{equation*} (f+g)(x)=f(x)+g(x) \end{equation*}

for all $f,g\in F$ and $x\in \R\text{.}$ We claim that $F$ is a group under pointwise addition. (For variety, in this proof we don't explicitly refer to $\G_1$–$\G_3\text{,}$ though we certainly do verify they hold.)

##### Example2.6.21

The set $F$ is not a group under function composition (do you see why?). But if we define $B$ to be the set of all bijections from $\R$ to $\R\text{,}$ then $B$ is a group under function composition. (Prove it!) $B$ is uncountably infinite and nonabelian.

##### Example2.6.22

Let $\langle G_1,*_1\rangle\text{,}$ $\langle G_2,*_2\rangle$ , $\ldots\text{,}$ $\langle G_n,*_n\rangle$ be groups ($n\in \Z^+$). Then the group product

\begin{equation*} G=G_1\times G_2\times \cdots \times G_n \end{equation*}

is a group under the componentwise operation $*$ defined by

\begin{equation*} (g_1,g_2,\ldots, g_n)*(h_1,h_2,\ldots,h_n)=(g_1*_1h_1, g_2*_2h_2,\ldots, g_n*_nh_n) \end{equation*}

for all $(g_1,g_2,\ldots, g_n),(h_1,h_2,\ldots,h_n)\in G\text{.}$

For instance, considering multiplication on $\R^*\text{,}$ matrix multiplication on $GL(2,\R)\text{,}$ and addition modulo $6$ on $\Z_6\text{,}$ we have that $\langle \R^*\times GL(2,\R) \times \Z_6,*\rangle$ is a group in which, for instance,

\begin{equation*} \left(-1, \begin{bmatrix} 1 \amp \phantom{-}3 \\ 0 \amp -1 \end{bmatrix}, 3\right) *\left(\pi, \begin{bmatrix} 2 \amp 1 \\ 1 \amp 1 \end{bmatrix}, 4\right)=\left(-\pi, \begin{bmatrix} \phantom{-}5 \amp \phantom{-}4 \\ -1 \amp -1 \end{bmatrix} ,1\right). \end{equation*}
##### Example2.6.23

A common example of a group product is the group $\Z_2^2\text{,}$ equipped with componentwise addition modulo 2.

##### Definition2.6.24

The group $\Z_2^2$ is known as the Klein 4-group. (Felix Klein was a German mathematician; you may have heard of him in relation to the Klein Bottle.) The group $\Z_2^2$ is sometimes denoted by $V\text{,}$ which stands for “Vierergruppe,” the German word for “four-group”.