Normal Complements of the Trivial Units in the Unit Group of some Integral Group Rings

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Let U(ZG) be the unit group of the integral group ring of a finite group G. It is well known that for abelian groups, the trivial units ±G have a free complement in U(ZG). For nonabelian groups, in [G. H. Cliff, S. K. Sehgal and A. R. Weiss, J. Algebra 73 (1981), no. 1, 167-185] it was shown that for some finite groups with a large abelian subgroup, there exists a torsion-free normal complement. Examples of such groups include the symmetric group S3 of degree 3 and the dihedral group D8 of order 8. In this paper the authors determine explicitly all normal complements of ±G in U(ZG), in case G = S3 or D8. It turns out that ±S3 has 4 normal complements in U(ZS3); three of them are free of rank 3, and one contains nontrivial torsion units of order 2. In the second case, they prove that ±D8 has precisely 8 normal complements in U(ZD8), all of them free of rank 3. Previous results were obtained in [P. J. Allen and C. R. Hobby, Proc. Amer. Math. Soc. 99 (1987), no. 1, 9-14] and [E. Jespers and G. Leal, Comm. Algebra 19 (1991), no. 6, 1809-1827].
Original languageEnglish
Pages (from-to)475-482
Number of pages8
JournalCommunications in Algebra
Publication statusPublished - 1 Jan 2003


  • unit group


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