# Section2.4Examples of groups/nongroups, Part I¶ permalink

Let's look at some examples of groups/nongroups.

##### Example2.4.1

We claim that $\Z$ is a group under addition. Indeed, $\langle\Z,+\rangle$ is a binary structure and that addition is associative on the integers. The integer $0$ acts as an identity element of $\Z$ under addition (since $a+0=0+a=a$ for each $a\in \Z$), and each element $a$ in $G$ has inverse $-a$ since $a+(-a)=-a+a=0\text{.}$

##### Example2.4.2

For each following binary structure $\langle G,*\rangle\text{,}$ determine whether or not $G$ is a group. For those that are not groups, determine the first group axiom that fails, and provide a proof that it fails.

1. $\langle \Q,+\rangle$

2. $\langle \Z,-\rangle$

3. $\langle \R,\cdot\rangle$

4. $\langle \C^*,\cdot\rangle$

5. $\langle \R,+\rangle$

6. $\langle \Z^+,+\rangle$

7. $\langle \Z^*,\cdot\rangle$

8. $\langle \M_n(\R),+\rangle$

9. $\langle \C,+\rangle$

10. $\langle \Z,\cdot\rangle$

11. $\langle \R^*,\cdot\rangle$

12. $\langle \M_n(\R),\cdot\rangle$

If you have taken linear algebra, you have also probably seen a collection of matrices that is a group under matrix multiplication.

##### Definition2.4.3

Recall that given a square matrix $A\text{,}$ the notation $\det A$ denotes the determinant of $A\text{.}$ Let

\begin{equation*} GL(n,\R)=\{M\in \M_n(\R):\det M \neq 0\} \end{equation*}

(that is, let $GL(n,\R)$ be the set of all invertible $n \times n$ matrices over $\R$) , and let

\begin{equation*} SL(n,\R)=\{M\in \M_n(\R):\det M =1\}\text{.} \end{equation*}

These subsets of $\M_n(\R)$ are called, respectively, the general and special linear groups of degree $n$ over $\R$.

Note that these definitions imply that these subsets of $\M_n(\R)$ are groups (under some operation). Sure enough, they are!

##### Remark2.4.6

Throughout this course, if we are discussing a set $GL(n,\R)$ or $SL(n,\R)\text{,}$ you should assume $n\in \Z^+\text{,}$ unless otherwise noted.

We end this section with a final example.

##### Example2.4.7

Define $*$ on $\Q^*$ by $a*b=(ab)/2$ for all $a,b\in \Q^*\text{.}$ Prove that $\langle \Q^*,*\rangle$ is a group.