Is it possible to create a Bilinear Function with Already Assigned "Multiplicative" Input Groups?
If you can find such an efficiently computable function (other than the trivial solution $\hat{e}( x, y ) = 1\ \ $ for all $x$, $y$), then you have shown that the decisional Diffie-Hellman problem is easy.  That is, given $g$, $g^a$, $g^b$, $g^c$, you can check whether

$$ab = c$$

simply by testing:

$$\hat{e}( g^a, g^b ) = \hat{e}( g, g^c)$$

The decisional Diffie-Hellman problem is believed to be hard in appropriately sized Multiplicative Cyclic groups, hence we believe that there is no such efficiently computable $\hat{e}$