24. If a and b are distinct group elements, prove that either a² + b² or a³ + b³.
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- 9. Find all homomorphic images of the octic group.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)19. a. Show that is isomorphic to , where the group operation in each of , and is addition. b. Show that is isomorphic to , where all group operations are addition.
- Find all homomorphic images of the quaternion group.38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.
- 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.Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.
- Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.True or False Label each of the following statements as either true or false. An element in a group may have more than one inverse.Prove that the set of all complex numbers that have absolute value forms a group with respect to multiplication.