I am following John B. Fraleigh -- A first course in abstract algebra. On page 90 of the 7th edition he say that
-- Decomposition of a cycle into products of transposition is possible since it is just the same as seeing that rearrangement of $n$ objects can he achieved by successively interchainging of pairs.
Now, I understand that rearrangement of $n$ objects can he achieved by successively interchainging pairs, but I cannot for the life of me see that this rationalizes cycle decomposition into compositions of transpositions.
Here are my attempts:
(1) Consider the cycle $(2,3,1) = (2,1)(2,3) = (2,1)\circ(2,3)$.
If I start with (1,2,3) and first transpose (2,3), and then (2,1) I get this:
Starting with : (1,2,3), then by transposing (2,3) gives (1,3,2), then by transposing (2,1) gives (2,3,1)
Which seemed to work in some sense. But the same algorithm fails for
(2) Consider the cycle $(1,3,5,4) = (1,4)\circ(1,5)\circ(1,3)$
Starting with : (1,3,4,5), then by transposing (1,3) gives (3,1,4,5), then by transposing (1,5) gives (3,5,4,1), then by transposing (1,4) gives (3,5,1,4)
But $(3,5,1,4) \neq (1,3,5,4)$, so the algorithm failed.
Can you help me to understand that the rearrangement of object by successive interchanges of pairs is the same as decomposing cycles into products of transpositions? A somewhat 'pictorial' explanation why these concepts are the same, would be best.