An odd group.

Since $dog^2 \neq dog$ , we note that $dog$ cannot be the identity element of the group. In particular, since $dog^2 = potato$ , then $dog \cdot potato = dog^3 = 1$ by Lagrange's theorem , so $dog$ and $potato$ are inverses of each other, and neither are the identity element.

The third element, $cat$ , must be the identity element. Thus, $cat \cdot dog = dog$ .

One can also reach this conclusion by process of elimination and a group multiplication table, but it is messier.

That $dog^2 = potato$ is compatible with a group structure can be seen by the group isomorphism to $\mathbb{Z}/3\mathbb{Z}$ given by $cat \mapsto 0, dog \mapsto 1,$ and $potato \mapsto 2$ . While the isomorphism class of the group with three elements is determined uniquely, the specific isomorphism is not: the other possible isomorphism could instead map $dog\mapsto 2$ and $potato \mapsto 1$ ; however, in both of these isomorphisms, $cat\mapsto 0$ .

Any group of $3$ elements is cyclic and hence isomorphic to $\Z_{3}$ . Where $\Z_{3}$ is the group under addition of integers modulo $3$ .

So the elements are $\{0,1,2\}$ . $1+1 = 2$ and $2+2 = 1$ . Hence $1$ is either the $\text{dog}$ or $\text{potato}$ and $2$ is also either $\text{dog}$ or $\text{potato}$ . Neither of $\text{dog}$ or $\text{potato}$ can be the identity element because there is no element of order $2$ in $\Z_{3}$ which implies that the square of a non-identity element is also a non identity element . hence $\text{cat}$ must be the identity element.

Hence $\text{cat+dog = dog}$

Nice solution already exists, but I just wanna post mine.

dog can't be identity element since $dog*dog\neq dog$ .

assume potato is identity element, then $potato*potato=potata$ , so dog and cat is self-inverse element but $cat*dog$ can't be cat because dog is not identity element, can't be dog with similar argument, and can't be potato because cat is not the inverse of dog. which is contradict because G is close under binary operation, so potato is not identity element

hence, cat is identity element so $cat*dog=dog$