Cube map is endomorphism implies class three

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Statement

Suppose G is a group such that the cube map x \mapsto x^3 is an endomorphism of G.

Then, G is a nilpotent group and its nilpotency class is at most three.

Facts used

  1. Levi's characterization of 3-abelian groups
  2. 2-Engel implies class three for groups

Proof

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