Let G be a group of order 80 and g be an element of G of order 20. Show that there is no element x of G satisfying 230 = g.
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- 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)Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.9. Suppose that and are subgroups of the abelian group such that . Prove that .6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.