UNITS IN ZS3 (AFTER MARCINIAK AND SEHGAL)

Summary. We render with a few minor variants the geometrical approach to the description of the group of units of the integral group ring ZS3 as given by Marciniak and Sehgal [Units in group rings and geometry, Methods in ring theory (Levico Terme, 1997)]. 1. Objective. In the literature one can nd quite a few explicit descriptions of the unit group of ZG for particular nite groups G. See [10] for references up to 1990, as well as the more recent survey [4] and the paper [8]. Of course, one may ask how an ultimate description of the group V(ZG) of augmentation one units should look like. The following program (cf. [9, Problem 17], [6], [8]) is ambitious: (a) Find a presentation for V(ZG); (b) identify the generators with a set of \natural " units in ZG; (c) nd an algorithm expressing any unit in V(ZG) as a product in the given generators. We remark that it is well known that V(ZG) is nitely presented; see the excellen