Prove that the group of real numbers (with addition as the binary operation) is isomorphic to the group of positive real numbers with respect to multiplication.

10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .

Find two groups of order 6 that are not isomorphic.

Find all subgroups of the quaternion group.

Exercises 35. Prove that any two groups of order are isomorphic.

Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.

Describe all subgroups of the group under addition.

15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.

17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of . (Thus the relation in Exercise does not have the symmetric property.) Exercise 15: 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.

Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.