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]