# Section5.3Exercises

##### 1

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ be a group with identity element $e\text{.}$

1. If $G$ is infinite and cyclic, then $G$ must have infinitely many generators.

2. There may be two distinct elements $a$ and $b$ of a group $G$ with $\langle a\rangle =\langle b\rangle\text{.}$

3. If $a,b\in G$ and $a\in \langle b\rangle$ then we must have $b\in \langle a\rangle\text{.}$

4. If $a\in G$ with $a^4=e\text{,}$ then $o(a)$ must equal $4\text{.}$

5. If $G$ is countable then $G$ must be cyclic.

##### 2

Give examples of the following.

1. An infinite noncyclic group $G$ containing an infinite cyclic subgroup $H\text{.}$

2. An infinite noncyclic group $G$ containing a finite nontrivial cyclic subgroup $H\text{.}$

3. A cyclic group $G$ containing exactly 20 elements.

4. A nontrivial cyclic group $G$ whose elements are all matrices.

5. A noncyclic group $G$ such that every proper subgroup of $G$ is cyclic.

##### 3

Find the orders of the following elements in the given groups.

1. $2\in \Z$

2. $-i\in \C^*$

3. $-I_2\in GL(2,\R)$

4. $-I_2\in \M_2(\R)$

5. $(6,8)\in \Z_{10}\times \Z_{10}$

##### 4

For each of the following, if the group is cyclic, list all of its generators. If the group is not cyclic, write NC.

1. $5\Z$

2. $\Z_{18}$

3. $\R$

4. $\langle \pi\rangle$ in $\R$

5. $\Z_2^2$

6. $\langle 8\rangle$ in $\Q^*$

##### 5

Explicitly identify the elements of the following subgroups of the given groups. You may use set-builder notation if the subgroup is infinite, or a conventional name for the subgroup if we have one.

1. $\langle 3\rangle$ in $\Z$

2. $\langle i\rangle$ in $C^*$

3. $\langle A\rangle\text{,}$ for $A=\left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 0 \end{array} \right]\in \M_2(\R)$

4. $\langle (2,3)\rangle$ in $\Z_4\times \Z_5$

5. $\langle B\rangle\text{,}$ for $B=\left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\in GL(2,\R)$

##### 6

Draw subgroup lattices for the following groups.

1. $\Z_6$

2. $\Z_{13}$

3. $\Z_{18}$

##### 7

Let $G$ be a group with no nontrivial proper subgroups. Prove that $G$ is cyclic.