Compute the orders of products of non-disjoint cycles:
(1 2 3)(3 2 4)
(1 2 3)(3 4 5)
I calcuated for a couple of times but still couldn't get the right answers.
Just write it as a premutation,
$\displaystyle \tau = \left( \begin{array}{cccc}1&2&3&4 \\ 2&4&3&1 \end{array} \right)$
Now find $\displaystyle \tau , \tau^2 , ... $
Until you find,
$\displaystyle \tau^n = \left( \begin{array}{cccc}1&2&3&4 \\ 1&2&3&4 \end{array} \right)$