Composition of permutations in cycle notation For $p=(1\ 4\ 3\ 2)$, find $p^2$. The textbook states that the solution is $p^2=(1\ 3)(2\ 4)$.
Now I understand that $1 \mapsto 4 \mapsto 3$, and $2 \mapsto 1 \mapsto 4$, however, why must the result be written as the composition of two cycles instead of as just one cycle?

Your work appears to be correct, but you're a little confused as to how to represent the answer. Let's work it out for all inputs:
$$ 1\mapsto4\mapsto3, \quad 2\mapsto1\mapsto4, \quad 3\mapsto2\mapsto1, \quad 4\mapsto3\mapsto2. $$
But then, in the end of the day, the intermediate steps don't matter, so let's state just the end results of applying $p^2$ to each of the inputs:
$$ 1\mapsto3, \quad 2\mapsto4, \quad 3\mapsto1, \quad 4\mapsto2. $$
We can clearly see that there are two separate cycles here: $1$ and $3$ are mapped into each other, and $2$ and $4$ are mapped into each other. So using the disjoint cycles notation, we have to write that $p^2=(1\;3)(2\;4)$.
To address a follow-up question from our conversation above in the comments, this not the same as $(1\;3\;2\;4)$ because they act differently on some inputs. For example, $(1\;3)(2\;4)$ maps $3$ to $1$, while $(1\;3\;2\;4)$ maps $3$ to $2$.
