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. #m/thm/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 $⟨ 𝑏 𝑘 ⟩ = ⟨ 𝑏 ⟩$ iff $g c d (𝑘, 𝑑) = 1$ . The number of such elements is exactly $𝜙 (𝑑)$ .

Significantly, there is no dependence on $𝑛$ , and hence $ℤ 73$ and $ℤ 8$ both have exactly $𝜙 (8) = 4$ elements of order 8.

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