## Exercises3.3Exercises

###### 1.

Prove or disprove: Every subgroup of the integers has finite index.

Hint

This is true for every proper nontrivial subgroup.

###### 2.

Prove or disprove: Every subgroup of the integers has finite order.

Hint

False.

###### 3.

Describe the left cosets of $SL_2( {\mathbb R} )$ in $GL_2( {\mathbb R})\text{.}$ What is the index of $SL_2( {\mathbb R} )$ in $GL_2( {\mathbb R})\text{?}$

###### 4.

Verify Euler's Theorem for $n = 15$ and $a = 4\text{.}$

Hint

$4^{\phi(15)} \equiv 4^8 \equiv 1 \pmod{15}\text{.}$

###### 5.

Use Fermat's Little Theorem to show that if $p = 4n + 3$ is prime, there is no solution to the equation $x^2 \equiv -1 \pmod{p}\text{.}$

###### 6.

Show that the integers have infinite index in the additive group of rational numbers.

###### 7.

Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.

###### 8.

Let $H$ be a subgroup of a group $G$ and suppose that $g_1, g_2 \in G\text{.}$ Prove that the following conditions are equivalent.

1. $\displaystyle g_1 H = g_2 H$

2. $\displaystyle H g_1^{-1} = H g_2^{-1}$

3. $\displaystyle g_1 H \subset g_2 H$

4. $\displaystyle g_2 \in g_1 H$

5. $\displaystyle g_1^{-1} g_2 \in H$

###### 9.

What fails in the proof of Theorem 3.8 if $\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H$ is defined by $\phi( gH ) = Hg\text{?}$

###### 10.

Suppose that $g^n = e\text{.}$ Show that the order of $g$ divides $n\text{.}$

###### 11.

The cycle structure of a permutation $\sigma$ is defined as the unordered list of the sizes of the cycles in the cycle decomposition $\sigma\text{.}$ For example, the permutation $\sigma = (12)(345)(78)(9)$ has cycle structure $(2,3,2,1)$ which can also be written as $(1, 2, 2, 3)\text{.}$

Show that any two permutations $\alpha, \beta \in S_n$ have the same cycle structure if and only if there exists a permutation $\gamma$ such that $\beta = \gamma \alpha \gamma^{-1}\text{.}$ If $\beta = \gamma \alpha \gamma^{-1}$ for some $\gamma \in S_n\text{,}$ then $\alpha$ and $\beta$ are conjugate.

###### 12.

If $|G| = 2n\text{,}$ prove that the number of elements of order $2$ is odd. Use this result to show that $G$ must contain a subgroup of order 2.

###### 13.

Let $H$ and $K$ be subgroups of a group $G\text{.}$ Prove that $gH \cap gK$ is a coset of $H \cap K$ in $G\text{.}$

Hint

Show that $g(H \cap K) = gH \cap gK\text{.}$

###### 14.

Let $H$ and $K$ be subgroups of a group $G\text{.}$ Define a relation $\sim$ on $G$ by $a \sim b$ if there exists an $h \in H$ and a $k \in K$ such that $hak = b\text{.}$ Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of $H = \{ (1),(123), (132) \}$ in $A_4\text{.}$

###### 15.

Let $G$ be a cyclic group of order $n\text{.}$ Show that there are exactly $\phi(n)$ generators for $G\text{.}$

###### 16.

Let $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\text{,}$ where $p_1, p_2, \ldots, p_k$ are distinct primes. Prove that

\begin{equation*} \phi(n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right)\cdots \left( 1 - \frac{1}{p_k} \right)\text{.} \end{equation*}
Hint

If $\gcd(m,n) = 1\text{,}$ then $\phi(mn) = \phi(m)\phi(n)$ ( Exercise 6.1.5.11).

###### 17.

Show that

\begin{equation*} n = \sum_{d \mid n} \phi(d) \end{equation*}

for all positive integers $n\text{.}$

###### 18.

For each of the following groups $G\text{,}$ determine whether $H$ is a normal subgroup of $G\text{.}$ If $H$ is a normal subgroup, write out a Cayley table for the factor group $G/H\text{.}$

1. $G = S_4$ and $H = A_4$

2. $G = A_5$ and $H = \{ (1), (123), (132) \}$

3. $G = S_4$ and $H = D_4$

4. $G = Q_8$ and $H = \{ 1, -1, I, -I \}$

5. $G = {\mathbb Z}$ and $H = 5 {\mathbb Z}$

Hint

(a)

\begin{equation*} \begin{array}{c|cc} & A_4 & (12)A_4 \\ \hline A_4 & A_4 & (12) A_4 \\ (12) A_4 & (12) A_4 & A_4 \end{array} \end{equation*}

(c) $D_4$ is not normal in $S_4\text{.}$

###### 19.

Find all the subgroups of $D_4\text{.}$ Which subgroups are normal? What are all the factor groups of $D_4$ up to isomorphism?

###### 20.

Find all the subgroups of the quaternion group, $Q_8\text{.}$ Which subgroups are normal? What are all the factor groups of $Q_8$ up to isomorphism?

###### 21.

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

\begin{equation*} \begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\text{,} \end{equation*}

where $a\text{,}$ $b\text{,}$ $c \in {\mathbb R}$ and $ac \neq 0\text{.}$ Let $U$ consist of matrices of the form

\begin{equation*} \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\text{,} \end{equation*}

where $x \in {\mathbb R}\text{.}$

1. Show that $U$ is a subgroup of $T\text{.}$

2. Prove that $U$ is abelian.

3. Prove that $U$ is normal in $T\text{.}$

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

5. Is $T$ normal in $GL_2( {\mathbb R})\text{?}$

###### 22.

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

###### 23.

If $G$ is abelian, prove that $G/H$ must also be abelian.

###### 24.

Prove or disprove: If $H$ is a normal subgroup of $G$ such that $H$ and $G/H$ are abelian, then $G$ is abelian.

###### 25.

If $G$ is cyclic, prove that $G/H$ must also be cyclic.

Hint

If $a \in G$ is a generator for $G\text{,}$ then $aH$ is a generator for $G/H\text{.}$

###### 26.

Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.

###### 27.

Let $H$ be a subgroup of index $2$ of a group $G\text{.}$ Prove that $H$ must be a normal subgroup of $G\text{.}$ Conclude that $S_n$ is not simple for $n \geq 3\text{.}$

###### 28.

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

Hint

For any $g \in G\text{,}$ show that the map $i_g : G \to G$ defined by $i_g : x \mapsto gxg^{-1}$ is an isomorphism of $G$ with itself. Then consider $i_g(H)\text{.}$

###### 29.

Define the centralizer of an element $g$ in a group $G$ to be the set

\begin{equation*} C(g) = \{ x \in G : xg = gx \}\text{.} \end{equation*}

Show that $C(g)$ is a subgroup of $G\text{.}$ If $g$ generates a normal subgroup of $G\text{,}$ prove that $C(g)$ is normal in $G\text{.}$

Hint

Suppose that $\langle g \rangle$ is normal in $G$ and let $y$ be an arbitrary element of $G\text{.}$ If $x \in C(g)\text{,}$ we must show that $y x y^{-1}$ is also in $C(g)\text{.}$ Show that $(y x y^{-1}) g = g (y x y^{-1})\text{.}$

###### 30.

Recall that the center of a group $G$ is the set

\begin{equation*} Z(G) = \{ x \in G : xg = gx \text{ for all } g \in G \}\text{.} \end{equation*}
1. Calculate the center of $S_3\text{.}$

2. Calculate the center of $GL_2 ( {\mathbb R} )\text{.}$

3. Show that the center of any group $G$ is a normal subgroup of $G\text{.}$

4. If $G / Z(G)$ is cyclic, show that $G$ is abelian.

###### 31.

Let $G$ be a group and let $G' = \langle aba^{- 1} b^{-1} \rangle\text{;}$ that is, $G'$ is the subgroup of all finite products of elements in $G$ of the form $aba^{-1}b^{-1}\text{.}$ The subgroup $G'$ is called the commutator subgroup of $G\text{.}$

1. Show that $G'$ is a normal subgroup of $G\text{.}$

2. Let $N$ be a normal subgroup of $G\text{.}$ Prove that $G/N$ is abelian if and only if $N$ contains the commutator subgroup of $G\text{.}$

Hint

(a) Let $g \in G$ and $h \in G'\text{.}$ If $h = aba^{-1}b^{-1}\text{,}$ then

\begin{align*} ghg^{-1} & = gaba^{-1}b^{-1}g^{-1}\\ & = (gag^{-1})(gbg^{-1})(ga^{-1}g^{-1})(gb^{-1}g^{-1})\\ & = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}\text{.} \end{align*}

We also need to show that if $h = h_1 \cdots h_n$ with $h_i = a_i b_i a_i^{-1} b_i^{-1}\text{,}$ then $ghg^{-1}$ is a product of elements of the same type. However, $ghg^{-1} = g h_1 \cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \cdots (gh_ng^{-1})\text{.}$