Edgar G. Goodaire

Loops whose loop rings in characteristic 2 are alternative

In this paper, the authors continue their investigation of loops which give rise to alternative loop rings. If the coefficient ring has characteristic 2, these loops turn out to form a surprisingly wide class, in contrast to the situation of characteristic ≠ 2. This paper describes many properties of this class, includes diverse examples of Moufang loops which are united by the fact that they have loop rings which are alternative, and discusses analogues in loop theory of a number of important group theoretic constructions.

Moufang loops with a unique nonidentity commutator (associator, square)

In [9], Griess considers a class of loops, called code loops, which have applicability to a construction of the monster and its nonassociative algebra (see [7]). These loops turn out to be Moufang loops which have a unique nonidentity commutator, a unique nonidentity associator, and a unique nonidentity square. Moufang loops with one or more of these properties also play an important role in the authors’ work on loops which have alternative loop rings [2-5, 81. For example, a nonassociative loop which has an alternative loop ring over a ring of characteristic different from two must be a Moufang loop with a unique nonidentity commutator and associator (which coincide). In this paper, we study Moufang loops with the properties in question. In Section 3, we concentrate on loops which have unique nonidentity commutators and/or associators and investigate some of their properties. In Section 4, we consider Moufang loops with a unique nonidentity square and show that these are exactly the code loops of Griess.

Units in alternative loop rings

In this paper, we show that certain well known theorems concerning units in integral group rings hold more generally for integral loop rings which are alternative.

A class of loops with right alternative loop rings

The purpose of this paper is to exhibit a class of loops which have strongly right alternative loop rings that are not alternative..

Torsion units in alternative loop rings

Let ZL denote the integral alternative loop ring of a finite loop L. If L is an abelian group, a well-known result of G. Higman says that ±g,g € L are the only torsion units (invertible elements of finite order) in ZL . When L is not abelian, another obvious source of units is the set ±y~l gy of conjugates of elements of L by invertible elements in the rational loop algebra QL . H. Zassenhaus has conjectured that all the torsion units in an integral group ring are of this form. In the alternative but not associative case, one can form potentially more torsion units by considering conjugates of conjugates V^\y7(g7l)V\ and so forth. In this paper we prove that every torsion unit in an alternative loop ring over Z is ± a conjugate of a conjugate of a loop element.

Isomorphisms of integral alternative loop rings

This paper first settles the “isomorphism problem” for alternative loop rings; namely, it is shown that a Moufang loop whose integral loop ring is alternative is determined up to isomorphism by that loop ring. Secondly, it is shown that every normalized automorphism of an alternative loop ringZL is the product of an inner automorphism ofQL and an authomorphism ofL.

Moufang unit loops torsion over their centres

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. We find necessary and sufficient conditions for the (Moufang) unit loop of RL to be torsion over its centre when R is the ring of rational integers or an arbitrary field. Over a field, torsion over the centre turns out to be equivalent to torsion of bounded exponent.

Involutions and Anticommutativity in Group Rings

Let g 7→ g∗ denote an involution on a group G. For any (commutative, associative) ring R (with 1), ∗ extends linearly to an involution of the group ring RG. An element α ∈ RG is symmetric if α∗ = α and skew-symmetric if α∗ = −α. The skew-symmetric elements are closed under the Lie bracket, [α, β] = αβ − βα. In this paper, we investigate when this set is also closed under the ring product in RG. The symmetric elements are closed under the Jordan product, α ◦ β = αβ + βα. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

Bol Loops of Nilpotence Class Two

Call a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

THREE-GENERATOR INDECOMPOSABLE RA LOOPS

ABSTRACT In [2], we showed that the minimally nonassociative RA loops (those which are not themselves associative but for which every proper subloop is associative) are precisely the RA loops which are indecomposable (that is, not nontrivial direct products) and which can be generated by three elements. Here, we investigate which RA loops have these two properties.