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.
Read more about this topic: Order (group Theory)
Famous quotes containing the words counting, order and/or elements:
“If you’re counting my eyebrows, I can help you. There are two.”
—Billy Wilder (b. 1906)
“In a drama of the highest order there is little food for censure or hatred; it teaches rather self-knowledge and self- respect.”
—Percy Bysshe Shelley (1792–1822)
“In verse one can take any damn constant one likes, one can alliterate, or assone, or rhyme, or quant, or smack, only one MUST leave the other elements irregular.”
—Ezra Pound (1885–1972)