Group order

Number of elements of order 𝑑 in a finite group

In a finite group , the number of elements of order is a multiple of , where 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 of order there exists a cyclic subgroup of order , which contains exactly generators, each of order . If there exists an element of order such that , then it too has a corresponding cyclic subgroup of order , which also contains exactly generators each of order , none of which may be contained in . Continuing in this fashion, it is clear that the number of elements in of order is where is some nonnegative integer.

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