Cyclic subgroup

Number of elements of each order in a cyclic group

Let ⟨𝑎⟩ be a cyclic subgroup of order 𝑛, and 𝑑 be a positive divisor of 𝑛. Then there exist exactly 𝜙(𝑛) elements in ⟨𝑎⟩ of order 𝑑, where 𝜙 is the Euler totient function. group

Proof

Let ⟨𝑏⟩ be the unique subgroup of order 𝑑 (guaranteed by the Fundamental theorem of cyclic groups). Then every element of order 𝑑 is a generator of ⟨𝑏⟩, and by Order of powers of a group element ⟨𝑏𝑘⟩ =⟨𝑏⟩ iff gcd(𝑘,𝑑) =1. The number of such elements is exactly 𝜙(𝑑).

Significantly, there is no dependence on 𝑛, and hence ℤ73 and ℤ8 both have exactly 𝜙(8) =4 elements of order 8.

---

tidy | en | SemBr