On the relation between group theory and loop theory

Abstract

Abstract Loops are nonassociative algebras (and are sometimes known as ‘nonassociative groups’) which can be investigated by using their multiplication groups. This connection to group theory is a source of several interesting group theoretical investigations and in the following four sections we try to cover some of the major problems in this area. Introduction Let Q be a groupoid with a neutral element e . If for any a, b ∈ Q each of the equations ax = b and ya = b has a unique solution, then we say that Q is a loop . For each a ∈ Q we have two permutations L a ( left translation ) and R a ( right translation ) on Q defined by L a ( x ) = ax and R a ( x ) = xa for every x ∈ Q . The permutation group M ( Q ) generated by the set of all left and right translations is called the multiplication group of Q . It is easy to see that M ( Q ) is transitive on Q and the stabilizers of elements of Q are conjugated in M ( Q ). The stabilizer of e ∈ Q is denoted by I ( Q ) and this stabilizer is called the inner mapping group of Q (it is interesting to observe that if Q is a group, then I ( Q ) is just the group of inner automorphisms of Q ). The concepts of the multiplication group and the inner mapping group of a loop were defined by Bruck [3] in 1946 in an article where he laid the foundation of loop theory.