4. Let G be the group of units modulo 40: G = U(50), and let H be the subgroup generated by 7: H = (7) = (7, 49, 43,1}. (a) (4 PTS) Find the index of H in G, [U (50): (7)]. (b) (4 PTS) Find the coset of H in G with representative 1 1, i.e. find 11(7). (c) (4 PTS) Since G is Abelian, the subgroup H must be a normal subgroup, and therefore we can form the factor group G/H. Find the order of the element 11(7) in this factor group. 5. Let G be the group Z25 X Z30- (a) (6 PTS) Find the number of elements in G with order 15. (b) (4 PTS) How many cyclic subgroups of order 15 does G have? (c) (4 PTS) Is G a cyclic group? Why or why not? R180 R270 H V D D O Ro Roo H V D D Ro Ro Rgo R180 R270 V Rgo R180 R270 Ro D D H Roo H D' D R180 R270 Ro Roo V R180 D D' V H R270 Ro Rgo R180 R270 R 180 R90 R270 H H D V D Ro R270 R90 V V D H D R180 Ro V D H R270 R90 Ro R180 D D R180 Ro H D V Rgo R270