Number of elements of order 𝑑 in a finite group

In a finite group $𝐺$ , the number of elements of order $𝑑$ is a multiple of $𝜙 (𝑑)$ , where $𝜙$ is the Euler totient function. #m/thm/group This is a corollary of the theorem on the Number of elements of each order in a cyclic group.

Proof

For each element $𝑎 \in 𝐺$ of order $𝑑$ there exists a cyclic subgroup $⟨ 𝑎 ⟩$ of order $𝑑$ , which contains exactly $𝜙 (𝑑)$ generators, each of order $𝑑$ . If there exists an element $𝑏 \in 𝐺$ of order $𝑑$ such that $𝑏 \notin ⟨ 𝑎 ⟩$ , then it too has a corresponding cyclic subgroup $⟨ 𝑏 ⟩$ of order $𝑑$ , which also contains exactly $𝜙 (𝑑)$ generators each of order $𝑑$ , none of which may be contained in $⟨ 𝑎 ⟩$ . Continuing in this fashion, it is clear that the number of elements in $𝐺$ of order $𝑑$ is $𝑛 𝜙 (𝑑)$ where $𝑛$ is some nonnegative integer.

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