# Kernel of a multihomomorphism implies completely divisibility-closed

## Statement

Suppose $G$ and $M$ are groups, $n \ge 2$ 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 $G$.

## Proof

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