Kernel of a multihomomorphism implies completely divisibility-closed

Statement

Suppose $G$ and $M$ are groups, $n\geq 2$ and:

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

(where the $G$ occurs $n$ times) is a multihomomorphism. Define:

$P:=\{x\in G\mid b(x,x_{2},x_{3},\dots ,x_{n}){\mbox{ is the identity element of }}M{\mbox{ for all }}x_{2},x_{3},\dots ,x_{n}\in G\}$

Then, $P$ is a completely divisibility-closed subgroup of $G$.

Facts used

1. Kernel of a multihomomorphism implies intersection of kernels of bihomomorphisms
2. Intersection of kernels of bihomomorphisms implies completely divisibility-closed

Proof

The proof follows by combining Facts (1) and (2).