Express the following permutations as products of transpositions and identify them as even or odd.

Question

So I am still getting the hang of cyclic notation.

Express the following permutations as products of transpositions and identify them as even or odd. I think this is saying express the following in pairs? like (xx), so my attempt:

a. (14356)=(61)(56)(35)(43)(14)

b. (156)(234)=(61)(56)(15)(42)(34)(23)

c. (1426)(142)=(61)(26)(42)(14)

My guess is that they're all even because I wrote them all as an even number of pairs? Not sure if this is even the correct reasoning or if my answers are right. Any guidance would be much appreciated!

Answer 1

Here are two simple things you might prove:

$$(a_1,a_2,...,a_n) = (a_1,a_2)(a_2,a_3)(a_3,a_4)...(a_{n-1},a_n)$$

$$(a_1,a_2,...,a_n) = (a_1,a_n)(a_1,a_{n-1})(a_1,a_{n-2})...(a_1,a_2)$$