How to show that this Cayley Table does not form a group

Given the following Cayley Table (where e is the identity element):

How would I go about proving that the table does not form a group?

I have checked closure, identity, inverses, and all 27 combinations of associativity excluding the ones that include the identity element.

1 Answer

With the translation $e=0$, $a=1$, $b=3$, and $c=2$, we can recognize that our table is the addition table modulo $4$. More formally, the structure $M$ with the given multiplication table is isomorphic to the additive group $\mathbb{Z}_4$, via the mapping $\varphi$ that takes $e$ to $0$, $a$ to $1$, $b$ to $3$, and $c$ to $2$. The fact that the table is a group table then follows from the standard fact that $\mathbb{Z}_4$ is a group.