If G is a group, then the set Z, defined by Z = {z € G: zx = xz \ x= G}. Prove that Z is a subgroup of G.
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- 45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )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)18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
- 34. Suppose that and are subgroups of the group . Prove that is a subgroup of .Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Label each of the following statements as either true or false. Let x,y, and z be elements of a group G. Then (xyz)1=x1y1z1.