(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]