## 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. $$H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}$$ 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 $$\sigma = (1 \, 2)(3 \, 4 \, 5)(7 \, 8)(9)$$ 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 $$H = \{ (1),(1 \, 2 \, 3), (1 \, 3 \, 2) \}$$ 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 $$n\text{.}$$