SOLUTION: I do not understand inverse permutations. For example, the permutation of (1 2 3 4 5), has an inverse of (1 5 4 3 2). How do we know this is true rather than some other permutation

Algebra ->  Permutations -> SOLUTION: I do not understand inverse permutations. For example, the permutation of (1 2 3 4 5), has an inverse of (1 5 4 3 2). How do we know this is true rather than some other permutation      Log On


   

Question 338155: I do not understand inverse permutations. For example, the permutation of (1 2 3 4 5), has an inverse of (1 5 4 3 2). How do we know this is true rather than some other permutation. Is there a valid, relatively simple equation to find the inverse of any permutation and does this imply that all permutations have an inverse?
Thank you,
Robert
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find the inverse of any permutation, simply write the elements in reverse order. In your case, the inverse of (1 2 3 4 5) is (5 4 3 2 1). It's perfectly possible to shift the elements, just as long as you do so in a consistent manner. So if we shift EVERY element to the right one place (the 1 will loop back to the beginning), we then get (1 5 4 3 2).


So (5 4 3 2 1) = (1 5 4 3 2) (ie they are the same permutation) is the inverse of (1 2 3 4 5)


It turns out that every permutation has an inverse. This is due to the fact that permutations form a group.