Consider the set Z₁0 under multiplication. [0] does not have a multiplicative inverse. So, Z10 is not a group under multiplication. Is Z₁0 - {[0]} a group under multiplication? (Hint: Think about the inverse axiom for a group.)
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- 9. Find all homomorphic images of the octic group.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.45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
- 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.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.