3.4: Exercises

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 $\mathbb{Z}\simeq \mathbb{Z}_n\text{.}$
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 $f:\mathbb{Z}\to\mathbb{Z}$ defined by $f(n)=2n\text{.}$
2. The function $g:\mathbb{R}\to\mathbb{R}$ defined by $g(x)=x^2\text{.}$
3. The function $h:\mathbb{Q}^*\to\mathbb{Q}^*$ defined by $h(x)=x^2\text{.}$

 

3. Define $d : GL(2,\mathbb{R})\to \mathbb{R}^*$ 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 $\mathbb{Z}_4$ and $\mathbb{Z}_8^{\times}\text{.}$ Use the group tables to decide whether or not $\mathbb{Z}_4$ and $\mathbb{Z}_8^{\times}$ are isomorphic to one another. (You do not need to provide a proof.)

5. Let $n\in \mathbb{Z}^+\text{.}$ Prove that $\langle n\mathbb{Z},+\rangle \simeq \langle \mathbb{Z},+\rangle\text{.}$
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 $\phi(a)^{-1}=\phi(a^{-1})\text{.}$