 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/10/b/solution

In a symmetric group, two permutations are in the same conjugacy class exactly when they have the same cycle type. For

S_{4}

the five possibilities and their sizes are

cycle type 1^{4} 2 1^{2} 2^{2} 314 representative e (12) (12) (34) (123) (1234) class size 16386 .

There are

(2 4) = 6

(3 4) (3 - 1)! = 8

(4 - 1)! = 6

four-cycles. Together with the identity these counts sum to

24 = ∣ S_{4} ∣

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