A characterization of the unit loop of the integral loop ring \({\mathbb{Z}}M_{16}(Q,2)\) (Q1209590)

The authors give a concrete description of the loop of units (invertible elements) \(U(\mathbb{Z} L)\) of the integral loop ring \(\mathbb{Z} L\), when \(L=M_{16}(Q,2)\) is the Moufang loop of order 16 that is not the loop of units of the Cayley numbers. In this case it is proved that \(U(\mathbb{Z} L)=\pm LV(\mathbb{Z} L)\), where \(V(\mathbb{Z} L)\) is a loop of integral matrices of determinant one in Zorn's rational vector matrix algebra. \(V(\mathbb{Z} L)\) is also a loop generated by three subgroups each isomorphic to the free group \(V(\mathbb{Z} D_ 4)\) of rank 3, where \(D_ 4\) is the dihedral group of order 8. The results of this work have been generalized and extended in the paper of \textit{E. G. Goodaire} [Can. J. Math. 44, 951-973 (1992)].