Group order

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. 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 ∈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 ∈𝐺 of order 𝑑 such that 𝑏 ∉⟨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 𝑛𝜙(𝑑) where 𝑛 is some nonnegative integer.

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