Cyclic subgroup

Number of elements of each order in a cyclic group

Let be a cyclic subgroup of order , and be a positive divisor of . Then there exist exactly elements in of order , where is the Euler totient function. group

Proof

Let be the unique subgroup of order (guaranteed by the Fundamental theorem of cyclic groups). Then every element of order is a generator of , and by Order of powers of a group element iff . The number of such elements is exactly .

Significantly, there is no dependence on , and hence and both have exactly elements of order 8.

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