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].
Dooms, A & Jespers, E 2003, 'Normal Complements of the Trivial Units in the Unit Group of some Integral Group Rings', Communications in Algebra, vol. 31, pp. 475-482. <http://www.informaworld.com/smpp/content~db=all?content=10.1081/AGB-120016770>
Dooms, A., & Jespers, E. (2003). Normal Complements of the Trivial Units in the Unit Group of some Integral Group Rings. Communications in Algebra, 31, 475-482. http://www.informaworld.com/smpp/content~db=all?content=10.1081/AGB-120016770
@article{7b52e24bbdd84e98992428b85988b640,
title = "Normal Complements of the Trivial Units in the Unit Group of some Integral Group Rings",
abstract = "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].",
keywords = "unit group",
author = "Ann Dooms and Eric Jespers",
year = "2003",
month = jan,
day = "1",
language = "English",
volume = "31",
pages = "475--482",
journal = "Communications in Algebra",
issn = "0092-7872",
publisher = "Taylor and Francis Ltd.",
}