$$\left. \begin{array} { l } { \text { Let } H = \{ a + b i | a , b \in \mathbf { R } , a ^ { 2 } + b ^ { 2 } = 1 \} . \text { Prove or disprove that } } \\ { H \text { is a subgroup of C's under multiplication. Describe the elements } } \\ { \text { of } H \text { geometrically. } } \end{array} \right.$$

$$\left. \begin{array} { l } { \text { Let } H = \{ A \in G L ( 2 , \mathbf { R } ) | \operatorname { det } A \text { is an integer power of } 2 \} . \text { Show that } } \\ { \text { His a subgroup of } G L ( 2 , \mathbf { R } ) .} \end{array} \right.$$

$$\begin{array} { l } { \text { Let } \mathbf { R } ^ { * } \text { denote the group of all nonzero real numbers under multi- } } \\ { \text { plication. Let } \mathbf { R } ^ { \text { - denote the group of positive real numbers under } } } \\ { \text { multiplication. Prove that } \mathbf { R } ^ { * } \text { is the internal direct product of } \mathbf { R } ^ { + } } \\ { \text { and the subgroup } \{ 1 , - 1 \} . } \end{array}$$

Show that the sequence $\left\{n^{4}\right\}$ diverges to infinity.

Compute the orders of the following groups. U(3), U(4), U(12), U(5), U(7), U(35), U(4), U(5), U(20), U(3), U(5), U(15). On the basis of your answers, make a conjecture about the relation- ship among |U(r)|, |U(s)|, and |U(rs)|

List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?

Suppose that $a$ and $b$ belong to a group and $a ^ { 5 } = e$ and $b ^ { 7 } = e$ . Write $a ^ { - 2 } b ^ { - 4 }$ and $\left( a ^ { 2 } b ^ { 4 } \right) ^ { - 2 }$ without using negative exponents.

$$\left. \begin{array} { c l } { \text { For each group in the following list, find the order of the group } } \\ { \text { and the order of each element in the group. What relation do you } } \\ { \text { see between the orders of the elements of a group and the order of } } \\ { \text { the group? } } \\ { } & { Z _ { 12 } , \quad U ( 10 ) , \quad U ( 12 ) , \quad U ( 20 ) , \quad D _ { 4 } } \end{array} \right.$$

$$\text { List the members of } H = \left\{ x ^ { 2 } | x \in D _ { 4 } \right\} \text { and } K = \left\{ x \in D _ { 4 } | x ^ { 2 } = e \right\}$$

$$\left. \begin{array} { l } { \text { Determine all elements of finite order in } R ^ { * } , \text { the group of nonzero } } \\ { \text { real numbers under multiplication. } } \end{array} \right.$$

$$\left. \begin{array} { l } { \text { Which of the following binary operations are commutative? } } \\ { \text { a. substraction of integers } } \\ { \text { b. division of nonzero real numbers } } \\ { \text { c. function composition of polynomials with real coefficients } } \\ { \text { d. multiplication of } 2 \times 2 \text { matrices with real entries } } \end{array} \right.$$

$$\left. \begin{array} { l } { \text { Which of the following sets are closed under the given operation? } } \\ { \mathbf { a } . \{ 0,4,8,12 \} \text { addition mod } 16 } \\ { \mathbf { b } . \{ 0,4,8,12 \} \text { addition mod } 15 } \\ { \mathbf { c } . \{ 1,4,7,13 \} \text { multiplication mod } 15 } \\ { \mathbf { d } . \{ 1,4,5,7 \} \text { multiplication mod } 9 } \end{array} \right.$$

$$\left. \begin{array} { l } { \text { Which of the following binary operations are associative? } } \\ { \text { a. multiplication mod } n } \\ { \text { b. division of nonzero rationals } } \\ { \text { c. function composition of polynomials with real coefficients } } \\ { \text { d. multiplication of } 2 \times 2 \text { matrices with integer entries } } \end{array} \right.$$

$$\left. \begin{array} { l } { \text { Which of the following binary operations are closed? } } \\ { \text { a. subtraction of positive integers } } \\ { \text { b. division of nonzero integers } } \\ { \text { c. function composition of polynomials with real coefficients } } \\ { \text { d. multiplication of } 2 \times 2 \text { matrices with integer entries } } \end{array} \right.$$