Counting By Order of Elements
Suppose G is a finite group of order n, and d is a divisor of n. The number of elements in G of order d is a multiple of φ(d), where φ is Euler's totient function, giving the number of positive integers no larger than d and coprime to it. For example in the case of S3, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite d such as d=6, since φ(6)=2, and there are zero elements of order 6 in S3.
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