Let G be a finite group, °IL(ZG) the group of units of the integral group ring ZG and %(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively %(ZG) for particular groups G. This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described °ti(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of

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... order 16 were discussed in their paper. Our main aim is to complete the description of °U{ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case. Note that there are 7 indecomposable non-abelian groups of order 16: (a,b\a 4 = l = b\ba = a 3 b), Q l6 = (a, b | a 8 = 1, a 4 = b 2 , ba = a 7 b), D= (a,b,c | a 2 = b 2 = c 4 = \,ac = ca, be = cb,ba = c 2 ab), D i6 =(a,b \a* = b 2 = l,ba = a 7 b), H={a,b\a 4 = b A = (ab) 2 = (a 2 , b) = 1). There are also 2 decomposable non-abelian groups of order 16: where Q is the quaternion group, C 2 the cyclic group of order 2 and D s =(a,b \a 4 = b 2 = l,ba = a 3 b),