Torsion units in alternative loop rings

Abstract:

Let ${\mathbf {Z}}L$ denote the integral alternative loop ring of a finite loop $L$. If $L$ is an abelian group, a well-known result of ${\text {G}}$. Higman says that $\pm g,g \in L$ are the only torsion units (invertible elements of finite order) in ${\mathbf {Z}}L$. When $L$ is not abelian, another obvious source of units is the set $\pm {\gamma ^{ - 1}}g\gamma$ of conjugates of elements of $L$ by invertible elements in the rational loop algebra ${\mathbf {Q}}L$. 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 $\gamma _{^1}^{ - 1}\left ( {\gamma _2^{ - 1}g{\gamma _2}} \right ){\gamma _1}$ and so forth. In this paper we prove that every torsion unit in an alternative loop ring over ${\mathbf {Z}}$ is $\pm$ a conjugate of a conjugate of a loop element.