Kernel of a bihomomorphism

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is a kernel of a bihomomorphism in $$G$$ if there exists a bihomomorphism:

$$b:G \times G \to M$$

for some group $$M$$, such that:

$$H = \{ x \in G \mid b(x,y) \mbox{ is the identity element of } M \}$$