<s>[INST] Two-row notation and disjoint cycles $$\pi = (12)(13)(826)(58)$$
I am asked to express $\pi$  as a product of disjoint cycles.
I am given the answer $(513268)$, why is it not $(513)(268)$
 [/INST] $\begin{array}{|c|c|c|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline (58)x & 1 & 2 & 3 & 4 & \color{blue}8 & 6 & 7 & \color{blue}5 \\
\hline (826)(58)x & 1 & \color{blue}6 & 3 & 4 & \color{blue}2 & \color{blue}8 & 7 & 5 \\
\hline (13)(826)(58)x & \color{blue}3 & 6 & \color{blue}1 & 4 & 2 & 8 & 7 & 5 \\
\hline (12)(13)(826)(58)x & 3 & 6 & \color{blue}2 & 4 & \color{blue}1 & 8 & 7 & 5 \\
\hline 
\end{array}$
Result : $1\to 3\to 2\to 6\to 8\to 5\to 1$ and $4$ and $7$ do not move.
$(12)(13)(826)(58)=(132685)$ which is the same as the stated $(513268)$.
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