Let $G$ and $H$ be sets , and $H$ has an abelian group structure ( for simplicity, operation is represented by $+$ and unit is $0$).
We know that $G$ has an operation $\cdot$ and the existence of the unit $e$.
Does the following claim hold in this situation?
Let $\phi : G \rightarrow H$ be a map . If $g_1 \cdot g_2 \cdot g_3 = e$ and $\phi(g_1)+\phi (g_2) + \phi(g_3) = 0$, then $\phi$ is "group" homomorphism.
In other words , will $G$ have a group structure in this situation?