Question

We consider the group Z₅₃. What are the possible element orders? How many elements exist for each order?

Answer 1

In the group Z₅₃, the possible element orders are 1, 2, 5, and 53. The number of elements for each order is 1 element of order 1, 1 element of order 2, 4 elements of order 5, and 52 elements of order 53.

Step-by-step explanation:

In the group Z₅₃, the possible element orders are 1, 2, 5, 53.

The number of elements for each order can be determined using a property of cyclic groups. In a cyclic group of order n, there is exactly one element of order d for every divisor d of n. Therefore, for the group Z₅₃, there is 1 element of order 1, 1 element of order 2, 4 elements of order 5, and 52 elements of order 53.