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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section2.4Examples of groups/nongroups, Part I

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


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 \(a+(-a)=-a+a=0\text{.}\)


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.


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!


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.


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