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It is shown that there are only four finite groups G with the property that G has a nonabelian free normal complement in the unit group of the integral group ZG. For finite commutative groups it is well known that the trivial units have a free abelian (normal) complement in the unit group of their integral group ring. Hence, it is natural to state the following problem: characterize the finite groups G which have a nonabelian free normal complement in the unit group of the integral group ring ZG. This question was posed to the author by M. M. Parmenter. In this paper it is shown that the only groups with this property are: 1. S3 = (a, b\a3 = 1, b2 = 1, ba = a~xb), the symmetric group of degree 3; 2. Z)4 = (a, b\a4 = 1, b2 = 1, ba = a~xb), the dihedral group of order 8; 3. T = (a, b\a6 = 1, b2 = a3, ba = a5b), the dicyclic group of order 12; and 4. P = (a, b\a4 = 1, b4 = 1, aba~xb~x = a2), a group of order 16. Note that in [4, 6, 9] it has been shown these groups indeed have the property under review. Throughout G denotes a finite group. In the first two lemmas we will make use of the following observation. Suppose A^ is a nonabelian free normal complement of the trivial units in %(ZG). Then, since %7(ZG)/N 3 ±G, in particular the factor group is finite, and because N is nonabelian free, it follows that %7(ZG) does not contain a free abelian subgroup of rank two. Lemma 1. If ±G has a nonabelian free normal complement in %7(ZG), then the order of an element in G is either 1,2, 3, 4, 5, 6, 8, or 12. Proof. Let g e G have order «, and let (g) denote the cyclic subgroup generated by g. Since í¿(Z(g)) does not contain a free abelian subgroup of rank 2, it follows from Higman's result (cf. [15, Theorem II.3.1]) that the rank r of Received by the editors December 10, 1992. 1991 Mathematics Subject Classification. Primary 16U60, 16S34. This work is supported in part by NSERC Grant OGP0036631, Canada. ©1994 American Mathematical Society 0002-9939/94 $1.00 +$.25 per page

Free normal complements and the unit group of integral group rings