Buchsteiner loops: Associators and constructions

Abstract

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that vertical bar Q vertical bar }= 32 if Q is not conjugacy closed.

We also show that vertical bar Q vertical bar }= 64 if there exists x. Q such that x(2) is not in the nucleus of x is an element of Q Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32.

Finally, we produce a Buchsteiner loop of order 128 that has both nilpotency class 3 and an abelian inner mapping group.

Keywords

• Buchsteiner loop
• conjugacy closed loop
• associator
• abelian inner mapping group