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A toy example would be this simple map with $\mathbb{G} = \mathbb{Z}/5$ to $\mathbb{G}_T = \mathbb{Z}^*/11$, as follows: $$e(x,y) = 3^{xy} \bmod 11$$ It's easy to verify that both equations hold (except that $e(0,0) = 1$; that's actually a necessary consequence of the first equation, and so I'll consider that an acceptable exception). Of course, even if ...