Find which of the operations given above has identity.

An element e ∈ Q will be the identity element for the operation * if

a * e = a = e * a, a ∈ Q.

We are given

$a^{*} b=\frac{a b}{4}$

$\Rightarrow a^{*} e=a$

$\Rightarrow \frac{a e}{4}=a$

$\Rightarrow e=4$

Similarly, it can be checked for $e^{*} a=a$, we get $e=4$

Thus, $e=4$ is the identity.