Structure of abelian quasi-groups

One of the most noticeable features of non-associative group-like systems is that the lack of associativity removes nearly all the power from the commutative law. For example, although many properties of groups are retained in systems which satisfy certain generalized associative laws, the addition of the commutative law does not usually reduce these systems to anything analogous to abelian groups. The properties which one usually associates with an abelian group, and which one would wish to retain in any non-associative generalization of this concept, are (a) Indices may be distributed: (ab)r = arbr. (b) All subgroups are normal. (c) The subgroups form a Dedekind structure. A fourth property which naturally comes to mind is that every abelian group is a direct product of cyclic groups. This, however, does not lend itself to generalization since the cyclic group itself loses all its simplicity when the associative law is relaxed. It will therefore play no part in the considerations of this paper. A definition of an abelian quasi-group which retains the above three properties has previously been given(1). It is a system closed under multiplication, which satisfies the quotient axiom and the generalized associative-commutative law (ab)(cd) = (ac)(bd). It is the purpose of this paper to give a complete account of the structure of these systems. The problem of constructing all abelian quasi-groups is solved in the sense that it is reduced to a group-theoretical problem. It is first shown, by consideration of the problem of extension, that every abelian quasi-group is a direct product of a self-unit quasi-group (one in which every element is a right unit) and one which contains an idempotent element. This latter type can always be constructed by performing certain transformations on the Cayley square of an abelian group, while the self-unit quasi-groups result from two applications of the same process. The results appear to indicate that the classic problems of extension, automorphisms, etc., although more cumbersome to handle, are not essentially more difficult than for abelian groups.