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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)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.
- Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then ab=ba.40. Find the commutator subgroup of each of the following groups. a. The quaternion group . b. The symmetric group .Construct a multiplication table for the group D5 of rigid motions of a regular pentagon with vertices 1,2,3,4,5.
- If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.Let G= a be a cyclic group of order 35. List all elements having each of the following orders in G. a. 2 b. 5 c. 7 d. 10