H and K are subgroups of a group G. By considering the four group axioms, prove that \(H \cap K\) is also a subgroup of G.

closure: let \(a,{\text{ }}b \in H \cap K\), so that \(a,{\text{ }}b \in H\) and \(a,{\text{ }}b \in K\)     M1

therefore \(ab \in H\) and \(ab \in K\) so that \(ab \in H \cap K\)     A1

associativity: this carries over from G     R1

identity: the identity \(e \in H\) and \(e \in K\)     M1

therefore \(e \in H \cap K\)     A1

inverse:

\(a \in H \cap K\) implies \(a \in H\) and \(a \in K\)     M1

it follows that \({a^{ - 1}} \in H\) and \({a^{ - 1}} \in K\)     A1

and therefore that \({a^{ - 1}} \in H \cap K\)     A1

the four group axioms are therefore satisfied     AG

[8 marks]

This question presented the most difficulty for students. Overall the candidates showed a lack of ability to present a formal proof. Some gained points for the proof of the identity element in the intersection and the statement that the associative property carries over from the group. However, the vast majority gained no points for the proof of closure or the inverse axioms.