With $\alpha = (12345)$ in the cycle notation, I should interpret it as:
$1\mapsto 2 \mapsto 3\mapsto 4\mapsto 5\mapsto 1$
I need to find out $\alpha^2$ and write it in cyclic notation. As I am not quite apt at it, I used the two row notation to solve it:
$\alpha = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{pmatrix}$
So $\alpha ^2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 1 & 2 \end{pmatrix}$
The answer provided is $\alpha^2 = (13524)$. Similarly, the answers to $\alpha^3$ etc were not matching. What am I doing wrong?