Intuition for expressing a cycle as a product of transpositions.I have recently learned the proof...

Willow Pratt

Willow Pratt

Answered

2022-07-02

Intuition for expressing a cycle as a product of transpositions.
I have recently learned the proof of:
( a 1   a 2     a n ) = ( a 1   a k ) ( a 1   a k 1 ) ( a 1   a 2 ) = ( a k   a k 1 ) ( a k   a k 2 ) ( a k   a 1 ) = ( a 1   a 2 ) ( a 2   a 3 ) ( a k 1   a k )
Now I can prove the above 3 equalities through induction but unfortunately induction, unlike a direct proof, doesn't quite tell me what is really going on. I'm struggling to understand this intuitively. I don't want to just memorize it.

Answer & Explanation

Kaylie Mcdonald

Kaylie Mcdonald

Expert

2022-07-03Added 19 answers

Step 1
The intuition you seem to need is how to read off from a permutation given as a product of cycles the mapping the permutation represents. If you consider a single cycle such as ( 1 2 3 ), then it is the permutation that maps 1 to 2, 2 to 3 and 3 to 1. When you look at a product of two or more cycles that may overlap, then you have to decide whether to work from left to right or from right to left. If I work from left to right ( 1 2 3 ) ( 2 4 ) maps 1 to 4 (via 2), 2 to 3, 3 to 1 and 4 to 2.
Step 2
So now it should be intuitively clear that ( 1 2 ) ( 2 3 ) ( 47 48 ) ( 48 1 ) (for example) is the cyclic permutation ( 1 2 48 ). And, in general that if the a i are all distinct that ( a 1 a 2 ) ( a 2 a 3 ) ( a n 1 a n ) ( a n a 1 ) is the cyclic permutation ( a 1 a 2 a n ).
Rapsinincke

Rapsinincke

Expert

2022-07-04Added 3 answers

Step 1
( a 1 , a 2 , , a n ) = ( a 1 , a n ) ( a 1 , a n 1 ) ( a 2 , a 1 )
Read the composition of cycles from right to left. How do they act on the element a 1 ? The right most transposition "sees" a 1 and takes it to a 2 .. None of the rest of the transpositions act on a 2 ..
Consider the element a k ,, the first several transpositions do not act on a k .. The first transposition that does is ( a 1 , a k ) .. That takes a k to a 1 .. The next transpositions takes that element to a k + 1 ..
Step 2
In this chain of transpositions, a 1 is acting as the interchange for all of the other elements. Every element gets sent to a 1 and then sent from a 1 to its destination.
( a n , a n 1 ) ( a n , a 1 ) is the same idea, except a k is acting as the interchange.
( a 1 , a 2 ) ( a 2 , a 3 ) ( a n 1 , a n ) even though it is ultimately the same cycle, the activity that is going on is a little bit different. Consider the element a k ,, it glides along unaffected by the first several transpositions, until it hits ( a k + 1 , a k ) gets shifted and then again the rest of the transpositions do nothing.

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