Say, an arbitrary permutation in $S_7$ as: $$\sigma = \begin{pmatrix} 1 2 3 4 5 6 7 \\3 2 5 7 6 1 4 \end{pmatrix} \implies (1356)(47)$$ Here, $\sigma^{-1}(3) = 1,\sigma^{-1}(4) = 7, \sigma(3) = 5, \sigma(4) = 7$.
To this we applied a transposition (swap among two elements) $\tau= (3,4)$. This lead us to get: $$\tau\sigma =\begin{pmatrix} 1 2 4 3 5 6 7 \\3 2 5 7 6 1 4\end{pmatrix} \implies (1374561)$$
But, the two disjoint cycles got merged.
Is it not wrong to affect the underlying structure?