math.GR — Group Theory

Bruck loops with abelian inner mapping groups: a short proof

Contributed by Bartosz Naskrecki

Phillips and Stanovský proved that a Bruck loop with abelian inner mapping group has central nilpotency class at most two and asked for a human-friendly proof. We give a short self-contained proof: a displacement cocycle produces a commutant-valued endomorphism, right alternativity for commutant elements forces its image to be fixed, and cocycle rigidity makes that image central. This yields the abelian center quotient.