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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?

jiten
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    What does "Is it not wrong to affect the underlying structure?" mean? Are you asking if it's really true that multiplying a permutation by a transposition changes its cycle type? Because yes, it does, almost always, and I really don't see a reason one might expect otherwise. – anon Jul 08 '22 at 00:24
  • @runway44 Yes, am referring to change in the cycle type. In graph theory, what is the term used ( for operation) to show such change in graph structure? – jiten Jul 08 '22 at 00:25

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