Homework 3
MATH 301/601
Test #3 is on Wednesday, March 12
 

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.

Prove that

$$G=\{a+b\sqrt{2}:a,b\in \mathbb{Q}\text{ and }a\text{ and }b\text{ are not both zero}\}$$

is a subgroup of ${ℝ}^{\times }$ under the group operation of multiplication.

*Exercise 2.

Let $H$ and $K$ be subgroup of a group $G$.

1. Prove that $H\cap K$ is a subgroup of $G$.

2. Prove or disprove: $H\cup K$ is a subgroup of $G$.

3. Prove that if $G$ is abelian, then $HK=\{hk:h\in H,k\in K\}$ is a subgroup of $G$.

Exercise 3.

Let $G$ be a group. The center of $G$ is the set

$$Z(G)=\{a\in G:ga=ag\text{ for all }g\in G\}.$$

Prove that $Z(G)$ is a subgroup of $G$.

Exercise 4.

1. Compute the center of $\mathrm{GL}(n,ℝ)$. Hint: Use the following test matrices: and .

2. Compute the center of $\mathrm{SL}(n,ℝ)$.

*Exercise 5.

Let $H$ be a subgroup of a group $G$. Define the relation $\sim$ on $G$ by $a\sim b$ if $b^{-1}a\in H$. Prove that $\sim$ is an equivalence relation on $G$.

**Exercise 6.

Suppose $H$ is a nonempty finite subset of a group $G$ and that $H$ is closed under multiplication (that is, $ab\in H$ for all $a,b\in H$). Prove that $H$ is a subgroup of $G$.

Exercise 7.

Let $G$ be a group, and let $a\in G$ have finite order.

1. Prove that the set $\{k\in \mathbb{N}:a^k=e\}$ is not empty.

2. Let $n\in \mathbb{N}$ be the least element of the set $\{k\in \mathbb{N}:a^k=e\}$ (which exists by part (a) and the well-ordering principle). Prove that if $i,j\in \{0,\mathrm{\dots },n-1\}$ such that $a^i=a^j$, then $i=j$.

3. Prove that $⟨a⟩=\{a^0,a^1,\mathrm{\dots },a^{n-1}\}$.

(Note: Parts (b) and (c) together show that $|a|=n$.)

Exercise 8.

Determine if $\mathbb{Q}$ is cyclic. Justify your answer.

*Exercise 9.

Let $A\in \mathrm{GL}(2,ℝ)$.

1. Prove that if $A$ has finite order, then $det(A)=\pm 1$.

2. Prove that if $det(A)=1$ and $|\mathrm{tr}(A)|>2$, then $A$ has a (real) eigenvalue $\lambda$ such that $|\lambda |\ne 1$, where $\mathrm{tr}(A)$ denotes the trace of $A$.

3. Let $A$ be as in part (b), so $det(A)=1$ and $|\mathrm{tr}(A)|>2$. Use the existence of an eigenvalue (and hence an eigenvector) to prove that if $det(A)=1$, then $A$ has infinite order.

Exercise 10.

Let $G$ be a group

1. Let $a,g\in G$. Prove that $|a|=|ga{g}^{-1}|$.

2. Let $a,b\in G$. Prove that $|ab|=|ba|$. (Hint: use part (a).)

Exercise 11.

Let $p$ be a prime number. Prove that ${\mathbb{Z}}_p$ has exactly two subgroups, namely the trivial subgroup and itself.

*Exercise 12.

Suppose $G$ is a nontrivial group in which the only two subgroups of $G$ are itself and the trivial subgroup.

1. Prove that $G$ is cyclic.

2. Using part (a), prove that $G$ is a finite group of prime order.

Exercise 13.

Let $p$ and $q$ be distinct prime numbers.

1. How many generators does ${\mathbb{Z}}_{pq}$ have? Justify your answer.

2. Let $r\in \mathbb{N}$. How many generators does ${\mathbb{Z}}_{p^r}$ have? Justify your answer.

*Exercise 14.

Let $a$ be an element of a group. For $n,m\in \mathbb{Z}$, find a generator for the group $⟨a^m⟩\cap ⟨a^n⟩$. Justify your answer.

Exercise 15.

Let $a$ and $b$ be elements in a group with relatively prime orders. Prove that $⟨a⟩\cap ⟨b⟩$ is the trivial subgroup.

**Exercise 16.

Let $p,q\in \mathbb{N}$ be relatively prime, and let $G$ be an abelian group of order $pq$. Prove that if $G$ contains elements of order $p$ and $q$, then $G$ is cyclic.

Exercise 17.

Let $G$ be an abelian group. Show that the elements of finite order in $G$ form a subgroup (known as the torsion subgroup).