Write (1,2,4,3) as a product of adjacent transpositions

Question

Write (1,2,4,3) as a product of adjacent transpositions. I am unsure how to do this. I wrote (1,2,4,3) as a product of transpositions as follows: (1,3)(1,4)(1,2).

Answer 1

Hint: $(x_1, . . . , x_m) = (x_1, x_m) \ldots (x_1, x_3)(x_1, x_2)$. So it seems fine.

Answer 2

Let's see what happens with your answer:

1 goes to 2, which then doesn't move after that. 2 goes to 1, which then goes to 4, which doesn't move after that. 4 goes to 1, which then goes to 3, and doesn't move after that. And finally, 3 goes to 1, so you have what you want.

You can think about these processes to generalise to a formula for decomposing cycles into transpositions.

Answer 3

Note that an adjacent transposition is defined as a transposition where the two elements are consecutive. Hence, $(1 2)$ is an adjacent transposition but not $(1 3)$.

Start by writing: $(1 2 4 3) = (1 3)(1 4)(1 2)$. Now, $(1 3)$ and $(1 4)$ are not adjacent. So, write: $$(1 3) = (1 2)(2 3)(2 1)$$ and $$(1 4) = [(2 3)(1 2)]^{-1}(3 4)[(2 3)(1 2)] = [(1 2)(2 3)](3 4)[(2 3)(1 2)]$$

Finally, $$(1 2 4 3) = (1 2)(2 3)(2 1)(1 2)(2 3)(3 4)(2 3)(1 2)(1 2) = (1 2)(3 4)(2 3)$$ is expressed as the required product.