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Topic: Permutation Groups
Replies: 13   Last Post: Feb 2, 2004 2:54 AM

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Chuck Cadman

Posts: 16
Registered: 12/12/04
Re: Permutation Groups
Posted: Oct 10, 1999 11:42 PM
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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.

Chuck Cadman

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