Expressing k-cycle as a product of transpositions

Question

Suppose $\pi = (a_1a_2a_3...a_k)$ is a $k$-cycle representing a permutation in $S_n$ for some $n \geq k$

I'm aware that $\pi$ can be expressed as a product of $k-1$ transpositions in the following way: $\pi = (a_1a_k)(a_1a_{k-1})...(a_1a_3)(a_1a_2)$

However I think that $\pi$ can also be expressed as the following product of $k-1$ transpositions:

$(a_1a_2)(a_2a_k)(a_2a_{k-1})...(a_2a_4)(a_2a_3)$

So I don't think the representation of $\pi$ as a product of transpositions is unique. If that's the case, does any representation of $\pi$ as a product of transpositions always contain $k-1$ transpositions or do different products of transpositions representing $\pi$ contain different numbers of transpositions? Is it a general rule that any $k$-cycle can be expressed only as a product of $k-1$ transpositions?

Answer 1

So I don't think the representation of π as a product of transpositions is unique.

This is correct, and you found an example indeed.

Is it a general rule that any $k$-cycle can be expressed only as a product of $k−1$ transpositions?

Well, this is not true since we can write $(a_1a_2) = (a_1a_2)(a_1a_2)(a_1a_2)$ for instance. What does not change here is the parity of the number of transpositions. But it is true that any $k$-cycle can be written as product of $k-1$ transpositions and we cannot write a $k$-cycle as product of less than $k-1$ transpositions (see here).