I have to find the number of elements in Aut(Z720) with order 6. Please suggest how to go about it. 1) Aut(Z720) isomorphic to U(720) (multiplicative group of units). 2 ) I am using the fundamental theorem of abelian group that a finite abelian group is isomorphic to the direct products of cyclic groups Zn. In this case, 720=16×9×5. Therefore, Aut(Z720)≅U(720)≅Z2×Z4×Z4×Z6. Now, the possible orders of elements in Z2:1,2; Z4:1,2,4; Z6:1,2,3,6. Using the result defining the order of an element in external direct products: If 6=Order(a,b,c,d)=lcm(Order(a),Order(b),Order(c),Order(d)) then: Case 1 : If Order(d)=6 then lcm(Order(a),Order(b),Order(c))=1 or 2. Using the the result for cyclic groups: for every divisor d of the order of a cyclic group G, there exists ϕ(d) elements in G with order d. It seems there are 16 elements. I am not sure though. Is this the correct way and how to proceed further? Please suggest.