A Characterization of the Unit Loop of the Integral Loop Ring Zm(16)(Q,2) Original Research Article

For a loop (group) G we denote by image(ZG) the loop (group) of units of the integral loop (group) ring ZG. Let L = M16(Q, 2) be the Moufang loop of order 16 that is not the loop of units of the Cayley numbers, and D4 the dihedral group of order 8. We prove that image(ZL) = ± LV(ZL), where V(ZL) is characterised as a loop of integral matrices of determinant one in Zorn′s rational vector matrix algebra. It follows that V(ZL) is generated by three subgroups each of which is isomorphic to the free group V(ZD4) of rank 3, and the latter is such that image(ZD4) = ± D4V(ZD4).