Find the order of each element in Z/10Z. Hence find all subgroups of Z/10Z. [Hint: everysubgroup of a cyclic group is cyclic.]

Step 1: Identify the elements in Z/10Z Z/10Z is the set of equivalence classes of integers modulo 10. This means that Z/10Z = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

Step 2: Find the order of each element The order of an element in a group is the smallest positive integer n such that a^n = e, where e is the identity element of the group. In Z/10Z, the identity element is 0.

• The order of 0 is 1 because 0*1 = 0.
• The order of 1 is 10 because 1*10 = 10 ≡ 0 (mod 10).
• The order of 2 is 5 because 2*5 = 10 ≡ 0 (mod 10).
• The order of 3 is 10 because 3*10 = 30 ≡ 0 (mod 10).
• The order of 4 is 5 because 4*5 = 20 ≡ 0 (mod 10).
• The order of 5 is 2 because 5*2 = 10 ≡ 0 (mod 10).
• The order of 6 is 5 because 6*5 = 30 ≡ 0 (mod 10).
• The order of 7 is 10 because 7*10 = 70 ≡ 0 (mod 10).
• The order of 8 is 5 because 8*5 = 40 ≡ 0 (mod 10).
• The order of 9 is 10 because 9*10 = 90 ≡ 0 (mod 10).

Step 3: Find all subgroups of Z/10Z Every subgroup of a cyclic group is cyclic. This means that the subgroups of Z/10Z are generated by the elements of Z/10Z.

• The subgroup generated by 0 is {0}.
• The subgroup generated by 1 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} = Z/10Z.
• The subgroup generated by 2 is {0, 2, 4, 6, 8}.
• The subgroup generated by 3 is {0, 3, 6, 9, 2, 5, 8, 1, 4, 7} = Z/10Z.
• The subgroup generated by 4 is {0, 4, 8, 2, 6}.
• The subgroup generated by 5 is {0, 5}.
• The subgroup generated by 6 is {0, 6, 2, 8, 4}.
• The subgroup generated by 7 is {0, 7, 4, 1, 8, 5, 2, 9, 6, 3} = Z/10Z.
• The subgroup generated by 8 is {0, 8, 6, 4, 2}.
• The subgroup generated by 9 is {0, 9, 8, 7, 6, 5, 4, 3, 2, 1} = Z/10Z.

So, the subgroups of Z/10Z are {0}, Z/10Z, {0, 2, 4, 6, 8}, {0, 5}.