What is the order of 30 in Z54Z Write out all of the element

What is the order of 30 in Z/54Z? Write out all of the elements and their orders in (30).

Solution

|30| = 54/gcd(54, 30) = 54/6 = 9.

Since H = 30 is a cyclic group of order 9. Hence the possible orders in H are 1, 3, 9.

If x = 30 (written multiplicatively),

Elements of order 9 : x = 30, x² = 6, x⁴ = 12, x⁵ = 42, x⁷ = 48, x⁸ = 24

Elements of order 3 : x³ = 36, x⁶= 18

Elements of order 1: 0 , and the trivial element.