Transpositions to work out odd/even permutations

Question

When working out the transpositions on simple permutations, e.g. a = (1 2 5 4 3) I know that a = (13)(14)(15)(12) so it is even

However I cant find transpositions of permutations that overlap, e.g say a = (1 2 3 4)(4 5 6)(2 5 3), I want to find out if this is odd or even but how do I write this as transpositions.

Answer 1

Write each cycle as a product of transpositions and count them. Or just imagine doing that and think about the parity of the total number, which will depend on the number of even length cycles you start with.

Answer 2

A cycle $(a_1 a_2 \cdots a_n)$ is odd iff $n$ is even.

A product of cycles is odd iff the number of odd cycles is odd.

(No need to assume that the cycles are disjoint.)