What is the arithmetic rule to merge a cycle and a non disjoint transposition?

Question

Assume you have an arbitrary cycle:

$(a_1,a_2,a_3\dots a_n)$

And a transposition $(a_i,a_j)$ for some $1 \leq i,j \leq n$

How can you merge the product $(a_1,a_2\dots a_n)(a_i,a_j)$?

And conversely the product$(a_i,a_j)(a_1,a_2\dots a_n)$?

EDIT:

And most importantly: $(a_i,a_j)(a_1,a_2\dots a_n)(a_i,a_j)$?

Answer 1

\begin{eqnarray*} (a_1 a_2 \cdots a_{i-1} a_i \cdots a_{j-1} a_j \cdots a_n) (a_i a_j) = (a_1 \cdots a_{i-1} a_i a_{j+1} \cdots a_n)(a_{i+1} \cdots a_{j-1} a_{j}) \end{eqnarray*} \begin{eqnarray*} (a_i a_j)(a_1 a_2 \cdots a_{i-1} a_i \cdots a_{j-1} a_j \cdots a_n) (a_i a_j) = (a_1 \cdots a_{i-1} a_j a_{i+1} \cdots a_{j-1} a_i a_{j+1} \cdots a_n) \end{eqnarray*} So $a_i$ and $a_j$ will change places.