Copy and complete the following Cayley table for $\{ T,{\text{ }} * \}$.

Prove that $\{ T,{\text{ }} * \}$ forms an Abelian group.

Find the order of each element in $T$.

Given that $\{ H,{\text{ }} * \}$ is the subgroup of $\{ T,{\text{ }} * \}$ of order $2$, partition $T$ into the left cosets with respect to $H$.

Cayley table is

     A4

award A4 for all 16 correct, A3 for up to 2 errors, A2 for up to 4 errors, A1 for up to 6 errors

[4 marks]

closed as no other element appears in the Cayley table     A1

symmetrical about the leading diagonal so commutative     R1

hence it is Abelian

$0$ is the identity

as $x * 0( = 0 * x) = x + 0 - 0 = x$     A1

$0$ and $2$ are self inverse, $3$ and $5$ is an inverse pair, $4$ and $6$ is an inverse pair     A1

Note:     Accept “Every row and every column has a $0$ so each element has an inverse”.

$(a * b) * c = (a + b - ab) * c = a + b - ab + c - (a + b - ab)c$     M1

$= a + b + c - ab - ac - bc + abc$     A1

$a * (b * c) = a * (b + c - bc) = a + b + c - bc - a(b + c - bc)$     A1

$= a + b + c - ab - ac - bc + abc$

so $(a * b) * c = a * (b * c)$ and $*$ is associative

Note:     Inclusion of mod 7 may be included at any stage.

[7 marks]

$0$ has order $1$ and $2$ has order $2$    A1

${3^2} = 4,{\text{ }}{3^3} = 2,{\text{ }}{3^4} = 6,{\text{ }}{3^5} = 5,{\text{ }}{3^6} = 0$ so $3$ has order $6$     A1

${4^2} = 6,{\text{ }}{4^3} = 0$ so $4$ has order $3$     A1

$5$ has order $6$ and $6$ has order $3$     A1

[4 marks]

$H = \{ 0,{\text{ }}2\}$     A1

$0 * \{ 0,{\text{ }}2\}  = \{ 0,{\text{ }}2\} ,{\text{ }}2 * \{ 0,{\text{ }}2\}  = \{ 2,{\text{ }}0\} ,{\text{ }}3 * \{ 0,{\text{ }}2\}  = \{ 3,{\text{ }}6\} ,{\text{ }}4 * \{ 0,{\text{ }}2\}  = \{ 4,{\text{ }}5\} ,$

$5 * \{ 0,{\text{ }}2\}  = \{ 5,{\text{ }}4\} ,{\text{ }}6 * \{ 0,{\text{ }}2\}  = \{ 6,{\text{ }}3\}$     M1

Note:     Award the M1 if sufficient examples are used to find at least two of the cosets.

so the left cosets are $\{ 0,{\text{ }}2\} ,{\text{ }}\{ 3,{\text{ }}6\} ,{\text{ }}\{ 4,{\text{ }}5\}$     A1

[3 marks]

Total [18 marks]