Kernel of a multihomomorphism implies completely divisibility-closed

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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.

Facts used

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


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