Question

Let p be a prime number, and let Z(pⁿ) be a subset of the abelian group Q/Z:

Z(pⁿ) = {[a/b] such that p does not divide b}
Determine the order of Z(pⁿ).

Answer 1

Z(pⁿ) is a subset of the abelian group Q/Z that contains elements such that p does not divide the denominator. The order of Z(pⁿ) is determined by the number of distinct elements that satisfy this condition.

Step-by-step explanation:

In this question, we are given the subset Z(pⁿ) of the abelian group Q/Z, where p is a prime number. Z(pⁿ) is defined as {[a/b] such that p does not divide b}.

The order of a group refers to the number of elements in that group. To determine the order of Z(pⁿ), we need to find how many distinct elements satisfy the given condition.

Let's consider an example where p = 2 and n = 3. In this case, Z(pⁿ) = {[a/b] such that 2 does not divide b}.

We can see that the elements of Z(pⁿ) are: {0, 1/1, 1/3, 2/1, 2/3}. Therefore, the order of Z(pⁿ) is 5.