Problem

Let \(\sigma\) be the permutation

$$ 1 \mapsto 13 \quad 2 \mapsto 2 \quad 3 \mapsto 15 \quad 4 \mapsto 14 \quad 5 \mapsto 10 \quad $$ $$ 6 \mapsto 6 \quad 7 \mapsto 12 \quad 8 \mapsto 3 \quad 9 \mapsto 4 \quad 10 \mapsto 1 \quad $$ $$ 11 \mapsto 7 \quad 12 \mapsto 9 \quad 13 \mapsto 5 \quad 14 \mapsto 11 \quad 15 \mapsto 8 \quad $$

and let \(\tau\) be the permutation

$$ 1 \mapsto 14 \quad 2 \mapsto 9 \quad 3 \mapsto 10 \quad 4 \mapsto 2 \quad 5 \mapsto 12 \quad $$ $$ 6 \mapsto 6 \quad 7 \mapsto 5 \quad 8 \mapsto 11 \quad 9 \mapsto 15 \quad 10 \mapsto 3 \quad $$ $$ 11 \mapsto 8 \quad 12 \mapsto 7 \quad 13 \mapsto 4 \quad 14 \mapsto 1 \quad 15 \mapsto 13. $$

Find the cycle decompositions of the following permutations: \(\sigma, \tau, \sigma^2, \sigma\tau, \tau\sigma\) and \(\tau^2\sigma\).

Solution

$$ \sigma = (1\ 13\ 5\ 10)(3\ 15\ 8)(4\ 14\ 11\ 7\ 12\ 9) $$

$$ \tau = (1\ 14)(2\ 9\ 15\ 13\ 4)(3\ 10)(5\ 12\ 7)(8\ 11) $$

$$ \sigma^2 = (1\ 5)(3\ 8\ 15)(4\ 11\ 12)(7\ 9\ 14)(10\ 13) $$

$$ \sigma\tau = (1\ 11\ 3)(2\ 4)(5\ 9\ 8\ 7\ 10\ 15)(13\ 14) $$

$$ \tau\sigma = (1\ 4)(2\ 9)(3\ 13\ 12\ 15\ 11\ 5)(8\ 10\ 14) $$

$$ \tau^2\sigma = (1\ 2\ 15\ 8\ 3\ 4\ 14\ 11\ 12\ 13\ 7\ 5\ 10) $$

$$\tag*{$\blacksquare$}$$