4 The elements \(a , b , c , d\) are combined according to the operation table below, to form a group \(G\) of order 4.
| \(a\) | \(b\) | \(c\) | \(d\) |
| \(a\) | \(b\) | \(a\) | \(d\) | \(c\) |
| \(b\) | \(a\) | \(b\) | \(c\) | \(d\) |
| \(c\) | \(d\) | \(c\) | \(a\) | \(b\) |
| \(d\) | \(c\) | \(d\) | \(b\) | \(a\) |
Group \(G\) is isomorphic either to the multiplicative group \(H = \left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\) or to the multiplicative group \(K = \{ e , p , q , p q \}\). It is given that \(r ^ { 4 } = e\) in group \(H\) and that \(p ^ { 2 } = q ^ { 2 } = e\) in group \(K\), where \(e\) denotes the identity in each group.
- Write down the operation tables for \(H\) and \(K\).
- State the identity element of \(G\).
- Demonstrate the isomorphism between \(G\) and either \(H\) or \(K\) by listing how the elements of \(G\) correspond to the elements of the other group. If the correspondence can be shown in more than one way, list the alternative correspondence(s).