Let $\{ G,{\text{ }} * \}$ be a finite group and let H be a non-empty subset of G . Prove that $\{ H,{\text{ }} * \}$ is a group if H is closed under $*$.

the associativity property carries over from G     R1

closure is given     R1

let $h \in H$ and let n denote the order of h, (this is finite because G is finite)     M1

it follows that ${h^n} = e$, the identity element     R1

and since H is closed, $e \in H$     R1

since $h * {h^{n - 1}} = e$     M1

it follows that ${h^{n - 1}}$ is the inverse, ${h^{ - 1}}$, of h     R1

and since H is closed, ${h^{ - 1}} \in H$ so each element of H has an inverse element     R1

the four requirements for H to be a group are therefore satisfied     AG

[8 marks]