Jack Patel wrote in message ... >I understand the theory of Permutaion Groups very well and understand >the concept of an alternating subgroup of degree n. > >However, I am stumped on these types of questions: > >1) How many elements of order 5 are in S7? (S=Symmetric Group) >2) How many odd permutations of order 4 does S6 have? >3) How many elements of order 5 are there in A6? (A=Alternating) >4) What is the maximum order of any element in A10? (I think 21 ?) > >I especially would like to know how to do number 1 because it is a >very basic problem.
Given the fact that any permutation can be expressed as a product of disjoint cycles, it's easy to show that the only elements of order 5 are 5-cycles (since 5 is prime). So you just need to count the number of ways that you can choose 5 distinct numbers between 1 and 7 and multiply by the number of unique 5-cycles that you can obtain from any such collection. I think it's 7!/10.