<s>[INST] The order of computing composition of permutation when using cycle notation Given two permutations $p,q$, say $p=(234)$ and $q=(123)$, we can compute the product permutation $pq$ by writing out $pq=(234)(123)$, then turn it into disjoint cycles.
In my understanding, we shall do the latter step in the backward direction: first evaluate on $q$, then on $p$. This is because the product permutation really means the composition of functions. And this gives $pq=(13)(24)$.
However, there are several times that I saw people on the internet doing this in the forward manner, which would give $pq=(12)(34)$.
Could anyone please clarify for me if I'm doing this correctly? Thanks in advance!
 [/INST] One can understand this by writing out the tables explicitly. Note that permutation $pq$ is applying $q$ and then applying $p$.
\begin{array}{|c|c|c|}
\hline
\text{permutation} & \text{acts on} & \text{produces}\\
\hline
pq=(234)(123)&1&3 \\
\hline
pq=(234)(123)&2&4 \\
\hline
pq=(234)(123)&3&1 \\
\hline
pq=(234)(123)&4&2 \\
\hline
\end{array}
The disjoint cycles are now visible:
\begin{array}{|c|c|c|}
\hline
\text{permutation} & \text{acts on} & \text{produces}\\
\hline
pq=(13)(24)&1&3 \\
\hline
pq=(13)(24)&2&4 \\
\hline
pq=(13)(24)&3&1 \\
\hline
pq=(13)(24)&4&2 \\
\hline
\end{array}
We will use the table below when told specifically that we apply permutation p and then q,
\begin{array}{|c|c|c|}
\hline
\text{permutation} & \text{acts on} & \text{produces}\\
\hline
qp=(123)(234)&1&2 \\
\hline
qp=(123)(234)&2&1 \\
\hline
qp=(123)(234)&3&4 \\
\hline
qp=(123)(234)&4&3 \\
\hline
\end{array}
\begin{array}{|c|c|c|}
\hline
\text{permutation} & \text{acts on} & \text{produces}\\
\hline
qp=(12)(34)&1&2 \\
\hline
qp=(12)(34)&2&1 \\
\hline
qp=(12)(34)&3&4 \\
\hline
qp=(12)(34)&4&3 \\
\hline
\end{array}
</s>