Elements of S n of Order Dividing a Given Number Joshua Fallon, Shanzhen Gao, Shaun Sullivan, Heinrich Niederhausen Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431 We present some formulas and properties for the elements of S n of order dividing a given number. 1 The Motivation For positive integers n and k, let F (n; k) be the number of those mappings of an nelement set into itself whose k-th iterate is the identity map (e.g. F (3; 2) = 4), and let the number F (4; 2) + F (8; 2) + F (8; 3) be nice and lucky and happy for you 1 . Solution: F (4; 2) + F (8; 2) + F (8; 3) = 10 + 764 + 1233 = 2007: 2 Some Denitions Let S n be the symmetric group on n elements and n; k; i; j be positive integers. Dene f (n; k)= fg 2 S n : g k =1g and F (n; k)=j f (n; k) j : Dene g(n; k)= fg 2 S n : o(g)= kg and G(n; k)=j g(n; k) j : Dene u(n; k)= f(x; y) 2 S n  S n :(xy) k =1g and U (n; k)=j u(n; k) j : Dene: u A (n; k; i; j )= f(x; y) 2 A  S n :(x i y j ) k =1;A is a subset of S n g and U A (n; k; i; j )=j u A (n; k; i; j ) j : Dene v A (n; k; i; j )= f(x; y) 2 A  S n : o(x i y j )= k;A is a subset of S n g and V A (n; k)=j v A (n; k; i; j ) j : Dene f (n; k; i)= fg 2 S n : g ik =1g and F (n; k; i)=j f (n; k; i) j : Dene g(n; k; i)= fg 2 S n : o(g i )= kg and G(n; k; i)=j g(n; k; i) j : Dene F (n; k; i; j )=jf(x; y) 2 S n  S n :(x i y j ) k =1gj. Dene G(n; k; i; j )=jf(x; y) 2 S n  S n : o(x i y j )= kgj : Dene P (n; k; i)=jfx 2 S n : o(x i )= kgj. 3 Main Results Theorem 1 Let p be a prime, then F (n; p)= bn=pc X i=0 n! p i  (n  p  i)!  i! : Example: F (3; 2) = 4;F (4; 2) = 10;F (5; 2) = 26;F (4; 3) = 9;F (5; 5) = 25: 1