TITLE: Need help understanding product of cycles
QUESTION [7 upvotes]: I need to understand how product of cycles work. The textbook i am referring gives explanation only for simple products and just answer for bigger ones. i would be very thankful if someone can explain me how to work around products like this : 
$(1532)(14)(35)$ 
some relevant link where i can read and understand will also work.
Aman

REPLY [16 votes]: Note that different authors define multiplication of permutations differently; it's either left-to-right or right-to-left.  The distinction is whether we define $f \circ g$ as $x \mapsto f(g(x))$ or $x \mapsto g(f(x))$.  Check with your notes/books as to which you should use.
Left-to-right
We can draw the points and where they map to as follows:

Then we just follow the paths to find
\begin{align*}
1 & \mapsto 3 \\
2 & \mapsto 4 \\
3 & \mapsto 2 \\
4 & \mapsto 1 \\
5 & \mapsto 5. \\
\end{align*}
We can convert this to cycle notation $(1324)(5)$, or $(1324)$ for short.
Right-to-left
The drawing now looks like:

Again, we just follow the paths to find
\begin{align*}
1 & \mapsto 4 \\
2 & \mapsto 1 \\
3 & \mapsto 3 \\
4 & \mapsto 5 \\
5 & \mapsto 2. \\
\end{align*}
We can convert this to cycle notation $(1452)(5)$, or $(1452)$ for short.