Homework 6
MATH 301/601
Test #6 is on Wednesday, May 7
 

Instructions. Read the Homework Guide to make sure you understand how to successfully complete the assignment. Click here for a pdf version of the homework.

Exercise 1.

Find five non-isomorphic groups of order 8. Justify why no two of them are isomorphic. (You will need to learn about the quaternion group, see Example 3.15 in the book.)

Exercise 2.

Let $G$ be a group of order 20. If $G$ has subgroups $H$ and $K$ of orders 4 and 5, respectively, such that $hk=kh$ for all $h\in H$ and $k\in K$, prove that $G$ is the internal direct product of $H$ and $K$.

*Exercise 3.

1. Prove or disprove: there is a noncyclic abelian group of order 51. (Do not use the classification of finite abelian groups.)

2. Prove or disprove: there is a noncyclic abelian group of order 52.

Exercise 4.

Prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.

*Exercise 5.

Recall that we can express each element of $D_6$ as a product of $r$ and $s$, where $r,s\in D_6$ satisfy $|r|=6$, $|s|=2$, and $sr=r^{-1}s$; in particular, we have

$$D_6=\{\mathrm{id},r,r^2,r^3,r^4,r^5,s,sr,s{r}^2,s{r}^3,s{r}^4,s{r}^5\}.$$

Let $H=⟨r^3⟩$ and let $K=\{\mathrm{id},r^2,r^4,s,s{r}^2,s{r}^4\}$.

1. Prove that $D_6$ is the internal direct product of $H$ and $K$.

2. Show that $K$ is isomorphic to $S_3$.

3. Deduce that $D_6\cong S_3\times {\mathbb{Z}}_2$.

*Exercise 6.

The goal of this exercise is to prove the folowing statement:

Every group of order four is isomoprhic to either ${\mathbb{Z}}_4$ or ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

Let $G$ be a group of order 4. If $G$ has an element of order 4, then it must be cyclic, and hence isomorphic to ${\mathbb{Z}}_4$. So, assume that $G$ does not have an element of order 4; the goal is now to prove that $G$ is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

1. Prove that every non-identity element of $G$ has order two.

2. Let $g,h\in G$ be distinct elements. Let $K=⟨g⟩$ and let $H=⟨h⟩$. Prove that $G$ is the internal direct product of $H$ and $K$.

3. Use the above to prove that $G\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

Exercise 7.

Let $G$ be a group, and let $H$ and $K$ be subgroups of $G$ such that $G$ is the internal direct product of $H$ and $K$. Define the function $\phi :H\times K\to G$ by $\phi (h,k)=hk$. Prove that $\phi$ is an isomorphism.

Exercise 8.

For each of the following groups, find all their subgroups, determine which are normal, and classify the corresponding factor groups up to isomorphism.

1. the dihedral group $D_4$.

2. the quaternion group $Q_8$.

*Exercise 9.

Let $T$ be the group of nonsingular upper triangular $2\times 2$ matrices with entries in $ℝ$, that is, matrices of the form

where $a,b,c\in ℝ$ and $ac\ne 0$. Let $U$ consist of the matrices of the form

where $x\in ℝ$.

1. Show that $U$ is a subgroup of $T$.

2. Prove that $U$ is abelian.

3. Prove that $U$ is normal in $T$.

4. Show that $T/U$ is abelian.

5. Is $T$ normal in $\mathrm{GL}(2,ℝ)$?

Exercise 10.

Prove that the intersection of two normal subgroups is a normal subgroup.

*Exercise 11.

If a group $G$ has exactly one subgroup $H$ of order $k$, prove that $H$ is normal in $G$.

**Exercise 12.

Let $G$ be a group. Given $a,b\in G$, define $[a,b]=ab{a}^{-1}{b}^{-1}$, the group element $[a,b]$ is called the commutator of $a$ and $b$. The commutator subgroup of $G$, denoted $[G,G]$, is the subgroup generated by the set of commutators, ie, it is the subgroup consisting of products of commutators.

1. Prove that $[G,G]$ is a normal subgroup of $G$.

2. Prove that $G/[G,G^{\prime }]$ is abelian.

3. Let $N$ be a normal subgroup of $G$. Prove that $G/N$ is abelian if and only if $[G,G]\subset N$.

(The group $G/[G,G]$ is called the abelianization of $G$.)

Exercise 13.

Let $\phi :G_1\to G_2$ be a homomorphism.

1. Prove that $\phi (G_1)=\{\phi (g):g\in G_1\}$ is a subgroup of $G_2$.

2. Prove that if $G_1$ is abelian, then $\phi (G_1)$ is abelian.

3. Prove that if $G_1$ is cyclic, then $\phi (G_1)$ is cyclic.

4. Prove that if $H$ is a subgroup of $G_2$, then ${\phi }^{-1}(H)=\{g\in G_1:\phi (g)\in H\}$ is a subgroup of $G_1$.

Exercise 14.

Let $G$ be a cyclic group and let $a$ be a generator of $G$. If ${\phi }_1,{\phi }_2:G\to H$ are homomorphisms such that ${\phi }_1(a)={\phi }_2(a)$, prove that ${\phi }_1={\phi }_2$.

Exercise 15.

1. Find all homomorphisms from $\mathbb{Z}$ to ${\mathbb{Z}}_6$.

2. Explain why there is no homomorphism from ${\mathbb{Z}}_6$ to ${\mathbb{Z}}_4$ that sends $\overline{1}$ in ${\mathbb{Z}}_6$ to $\overline{1}$ in ${\mathbb{Z}}_4$.

3. Find all the homomorphisms from ${\mathbb{Z}}_{24}$ to ${\mathbb{Z}}_{18}$.

Definition. The kernel of a homomorphism $\phi :G\to H$, denoted $\ker \phi$, is defined by $\ker \phi =\{g\in G:\phi (g)=e_H\}$.

*Exercise 16.

Let $\phi :G\to H$ be a homomorphism. Prove that $\ker \phi$ is a normal subgroup of $G$.

*Exercise 17.

Prove that homomorphism $\phi :G\to H$ is injective if and only if $\ker \phi =\{e_G\}$.