(i)     Write the permutation $\alpha = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&4&6&2&1&5 \end{array}} \right)$ as a composition of disjoint cycles.

(ii)     State the order of $\alpha$.

(i)     Write the permutation $\beta = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 6&4&3&5&1&2 \end{array}} \right)$ as a composition of disjoint cycles.

(ii)     State the order of $\beta$.

Write the permutation $\alpha  \circ \beta$ as a composition of disjoint cycles.

Write the permutation $\beta  \circ \alpha$ as a composition of disjoint cycles.

State the order of $\{ G,{\text{ }} \circ \}$.

Find the number of permutations in $\{ G,{\text{ }} \circ \}$ which will result in $A$, $B$ and $A \cap B$ remaining unchanged.

(i)     $(1{\text{ }}3{\text{ }}6{\text{ }}5)(2{\text{ }}4)$     A1A1

(ii)     4     A1

Note: In (b) (c) and (d) single cycles can be omitted.

[3 marks]

(i)     $(1{\text{ }}6{\text{ }}2{\text{ }}4{\text{ }}5)(3)$     A1

(ii)     5     A1

[2 marks]

$\left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 5&2&6&1&3&4 \end{array}} \right) = (1{\text{ }}5{\text{ }}3{\text{ }}6{\text{ }}4)(2)$    (M1)A1

[2 marks]

$\left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 3&5&2&4&6&1 \end{array}} \right) = (1{\text{ }}3{\text{ }}2{\text{ }}5{\text{ }}6)(4)$    (M1)A1

Note: Award A2A0 for (c) and (d) combined, if answers are the wrong way round.

[2 marks]

$6! = 720$    A2

[2 marks]

any composition of the cycles (1 2), (3 4) and (5 6)     (M1)

so ${2^3} = 8$     A1

[2 marks]