Expressing Transpositions as Products of Adjacent Transpositions

[murps232]

Homework Statement

Show that every transposition (i,j)(1≤i≤j≤n) in S_n is expressible as a product of adjacent transpositions.

Also express the transposition (1,9) as a product of adjacent transpositions.

Homework Equations

none

The Attempt at a Solution

Really struggling to even start the proof.

Is the transposition (1,9)=(1,2)(2,3)(3,4)(4,5)(5,6)(6,7)(7,8)(8,9)?

 

[CompuChip]

I assume (i, j) means swapping i and j. You can check your answer if you have 9 small pieces of paper. That should also lead you on to a proof.

You are definitely on the right way that you have to "string" i through i + 1, i + 2, ... until it reaches j and vice versa.

 

[Stephen Tashi]

I suggest you start by working a specific example. Express (1,3) as the product of "adjacent transpositions". By the way, what is the definition of an "adjacent transposition"? A cycle like (1,2) ?