# Section3.4Exercises

##### 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 $G$ and $G'$ be groups.

1. If there exists a homomorphism $\phi\,:\,G\to G'\text{,}$ then $G$ and $G'$ must be isomorphic groups.

2. There is an integer $n\geq 2$ such that $\Z\simeq \Z_n\text{.}$

3. If $|G|=|G'|=3\text{,}$ then we must have $G\simeq G'\text{.}$

4. If $|G|=|G'|=4\text{,}$ then we must have $G\simeq G'\text{.}$

##### 2

For each of the following functions, prove or disprove that the function is (i) a homomorphism; (ii) an isomorphism. (Remember to work with the default operation on each of these groups!)

1. The function $f:\Z\to\Z$ defined by $f(n)=2n\text{.}$

2. The function $g:\R\to\R$ defined by $g(x)=x^2\text{.}$

3. The function $h:\Q^*\to\Q^*$ defined by $h(x)=x^2\text{.}$

##### 3

LDefine $d : GL(2,\R)\to \R^*$ by $d(A)=\det A\text{.}$ Prove/disprove that $d$ is:

1. a homomorphism

2. 1-1

3. onto

4. an isomorphism.

##### 4

Complete the group tables for $\Z_4$ and $\Z_8^{\times}\text{.}$ Use the group tables to decide whether or not $\Z_4$ and $\Z_8^{\times}$ are isomorphic to one another. (You do not need to provide a proof.)

##### 5

Let $n\in \Z^+\text{.}$ Prove that $\langle n\Z,+\rangle \simeq \langle \Z,+\rangle\text{.}$

##### 6

1. Let $G$ and $G'$ be groups, where $G$ is abelian and $G\simeq G'\text{.}$ Prove that $G'$ is abelian.

2. Give an example of groups $G$ and $G'\text{,}$ where $G$ is abelian and there exists a homomorphism from $G$ to $G'\text{,}$ but $G'$ is NOT abelian.

##### 7

Let $\langle G,\cdot\rangle$ and $\langle G',\cdot'\rangle$ be groups with identity elements $e$ and $e'\text{,}$ respectively, and let $\phi$ be a homomorphism from $G$ to $G'\text{.}$ Let $a\in G\text{.}$ Prove that $\phi(a)^{-1}=\phi(a^{-1})\text{.}$