4 The set \(L\) consists of all points \(( x , y )\) in the cartesian plane, with \(x \neq 0\). The operation ◇ is defined by \(( a , b ) \diamond ( c , d ) = ( a c , b + a d )\) for \(( a , b ) , ( c , d ) \in L\).
- Show that \(L\) is closed under ◇.
- Prove that \(\diamond\) is associative on \(L\).
- Find the identity element of \(L\) under ◇ .
- Find the inverse element of \(( a , b )\) under ◇.
- Find a subgroup of \(( L , \diamond )\) of order 2.