Date: Oct 10, 1999 11:42 PM
Author: Chuck Cadman
Subject: Re: Permutation Groups
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.