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

State the identity element in $\{ G,{\text{ }} \circ \}$.

Find

(i)     $p \circ p$;

(ii)     the inverse of $p \circ p$.

(i)     Find the maximum possible order of an element in $\{ H,{\text{ }} \circ \}$.

(ii)     Give an example of an element with this order.

the order of $(G,{\text{ }} \circ )$ is ${\text{lcm}}(6,{\text{ }}4)$     (M1)

$= 12$     A1

[2 marks]

$\left( 1 \right){\rm{ }}\left( 2 \right){\rm{ }}\left( 3 \right){\rm{ }}\left( 4 \right){\rm{ }}\left( 5 \right){\rm{ }}\left( 6 \right){\rm{ }}\left( 7 \right){\rm{ }}\left( 8 \right){\rm{ }}\left( 9 \right){\rm{ }}\left( {10} \right)$     A1

Note:     Accept ( ) or a word description.

[1 mark]

(i)     $p \circ p = (1{\text{ }}3{\text{ }}5)(2{\text{ }}4{\text{ }}6)(7{\text{ }}9)(810)$     (M1)A1

(ii)     its inverse $= (1{\text{ }}5{\text{ }}3)(2{\text{ }}6{\text{ }}4)(7{\text{ }}9)(810)$     A1A1

Note:     Award A1 for cycles of 2, A1 for cycles of 3.

[4 marks]

(i)     considering LCM of length of cycles with length $2$, $3$ and $5$     (M1)

$30$     A1

(ii)     eg$\;\;\;(1{\text{ }}2)(3{\text{ }}4{\text{ }}5)(6{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}10)$     A1

Note:     allow FT as long as the length of cycles adds to $10$ and their LCM is consistent with answer to part (i).

Note: Accept alternative notation for each part

[3 marks]

Total [10 marks]