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Inverse, Product of Permutations

Date: 04/27/2002 at 09:37:54
From: Meena Varma
Subject: The inverse and product of permutations

Hello Dr.Math,

I know how to calculate the signum of permutations, but I don't 
understand how to calculate the inverse or the product of 
permutations.

Could you please solve for:

p1 = |1 2 3 4 5 6| and p2 = |1 2 3 4 5 6|
     |2 3 1 5 4 6|          |1 2 3 6 4 5|
    
Thanking you in anticipation.

Regards,
Meena


Date: 04/27/2002 at 10:42:06
From: Doctor Mitteldorf
Subject: Re: The inverse and product of permutations

Dear Meena,

The inverse of a permutation is the permutation that "undoes" it.  
If p1 takes 1 into 2, then the inverse of p1 must take 2 into 1. You 
can write the inverse of the permutation you've listed as p1 by 
putting the top line on the bottom and the bottom line on top. Then, 
for convention's sake, you should rearrange the columns to be in 
standard order, starting with 

|1 2 3 4 5 6|  
|3 1              etc.  - you fill it in.


To calculate the product you must follow the chain of events:  
             ______________
            /              \   
p1 takes 1->2 and p2 takes 2->2, so the product takes 1->2.
p1 takes 2->3 and p2 takes 3->3, so the product takes 2->3.
p1 takes 3->1 and p2 takes 1->1, so the product takes 3->1.
p1 takes 4->5 and p2 takes 5->4, so the product takes 4->4.
...etc.

(Notice that the product p1*p2 is not the same as p2*p1. The product 
that we've calculated above is the one that is conventionally written 
p2*p1, with p1 acting first, then p2.)

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 04/27/2002 at 11:04:39
From: Meena Varma
Subject: The inverse  and product of permutations

Dear Dr.Mitteldorf,

That was an excellent explanation and a very expeditious reply! Now I
can do my university level Linear Algebra homework question :) This 
was the first question I ever asked here and I'm so delighted with the 
result!

Thank you so much!

Regards,
Meena
Associated Topics:
College Discrete Math
High School Discrete Mathematics
High School Permutations and Combinations

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