Why DDH is easy if a pairing function exists? When I look at some demos which demonstrate that DDH is easy if a pairing function $e$ exists, I usually see if $e(g,g^c)=e(g^a,g^b)$. Then we can know that $g^c=g^{ab}$, but why is this so? How do we remove the $e$? Let's imagine this is a symmetric-key cypher problem and $e:G_1 \times G_1 \rightarrow G_2$.

For example: here.
You don't 'remove the e', instead, you use the mathematical properties of it.

e satistifies the identities:

$$e(a^x, b) = e(a, b)^x$$

$$e(a, b^y) = e(a, b)^y$$

for any $a, b, x, y$

We also assume that the pairing is nontrivial (that is, $e(g,g) \ne 1$); typically, we further assume that $g$ generates a prime order group, this implies that $e(g,g)$ generates a group of the same order.

So, we have $e(g, g^c) = e(g, g)^c$ (second identity).

We also have $e(g^a, g^b) = e(g, g^b)^a = e(g, g)^{ab}$ (first and second identities).

So, if we have $e(g, g)^c = e(g, g)^{ab}$, because we know that the group generated by $e(g, g)$ and $g$ are isomorphic (even if we cannot compute that isomorphism in the $e(g,g)$ to $g$ direction; that's because all prime order groups of the same prime order are isomorphic), we then have $g^c = g^{ab}$