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## Exercises6.5Exercises

###### 1.

Suppose that $G$ is a finite group with an element $g$ of order $5$ and an element $h$ of order $7\text{.}$ Why must $|G| \geq 35\text{?}$

###### 2.

Suppose that $G$ is a finite group with $60$ elements. What are the orders of possible subgroups of $G\text{?}$

###### 3.

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

###### 4.

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

###### 5.

List the left and right cosets of the subgroups in each of the following.

1. $\langle 8 \rangle$ in ${\mathbb Z}_{24}$

2. $\langle 3 \rangle$ in $U(8)$

3. $3 {\mathbb Z}$ in ${\mathbb Z}$

4. $A_4$ in $S_4$

5. $A_n$ in $S_n$

6. $D_4$ in $S_4$

7. ${\mathbb T}$ in ${\mathbb C}^\ast$

8. $H = \{ (1), (123), (132) \}$ in $S_4$

###### 6.

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 $GL_2( {\mathbb R})\text{?}$

###### 7.

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

###### 8.

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

###### 9.

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

###### 10.

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

###### 11.

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

###### 12.

If $ghg^{-1} \in H$ for all $g \in G$ and $h \in H\text{,}$ show that right cosets are identical to left cosets. That is, show that $gH = Hg$ for all $g \in G\text{.}$

###### 13.

What fails in the proof of Theorem 6.8 if $\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H$ is defined by $\phi( gH ) = Hg\text{?}$

###### 14.

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

###### 15.

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.

###### 16.

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.

###### 17.

Suppose that $[G : H] = 2\text{.}$ If $a$ and $b$ are not in $H\text{,}$ show that $ab \in H\text{.}$

###### 18.

If $[G : H] = 2\text{,}$ prove that $gH = Hg\text{.}$

###### 19.

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

###### 20.

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 $A_4\text{.}$

###### 21.

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

###### 22.

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*}
###### 23.

Show that

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

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