Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Subnormal Subgroups of Group Ring Units

Author(s): Zbigniew S. Marciniak; Sudarshan K. Sehgal
Journal: Proc. Amer. Math. Soc. 126 (1998), 343-348.
MSC (1991): Primary 16S34, 16U60
MathSciNet review: 1423318
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Abstract: Let $G$ be an arbitrary group. If $a\in \mathbb{Z}G$ satisfies $a^{2}=0$ , $a\ne 0$ , then the units $1+a$ , $1+a^{*}$ generate a nonabelian free subgroup of units. As an application we show that if $G$ is contained in an almost subnormal subgroup $V$ of units in $\mathbb{Z}G$ then either $V$ contains a nonabelian free subgroup or all finite subgroups of $G$ are normal. This was known before to be true for finite groups $G$ only.

References:

1.: Ambrose, W., Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), 364-386. MR 7:126c
2.: Goncalves, J., Ritter, J., Sehgal, S.K., Subnormal subgroups in ${\mathcal U}_1{\mathbb Z}G$ , Proc. Amer. Math. Soc. 103 (1988), 375-382. MR 89h:20006
3.: Goncalves, J., Passman, D.S., Construction of free groups in the group of units of modular group algebras, Comm. Algebra 24 (1996), no. 13, 4211-4215. MR 97g:16043
4.: Hartley, B., Pickel, P.F., Free subgroups in the unit groups of integral group rings, Canadian Journal of Math. 32 (1980), 1342-1352. MR 82i:20008
5.: Jespers, E., Normal complements and the unit group of integral group rings, Proceedings of AMS 122 (1994), 59-66. MR 94k:16058
6.: Jespers, E., Leal, G., del Río, A., Products of free groups in the unit group of integral group rings, to appear in J.Algebra.
7.: Leal, G., del Río, A., Products of free groups in the unit group of integral group rings II, to appear in J. Algebra.
8.: Kargapolov, M., Mierzljakov, Yu., Fundamentals of the theory of groups, Springer Verlag, 1979. MR 80k:20002
9.: Lang, S., Algebra, Addison-Wesley Student Series, second printing 1970. MR 33:5416
10.: Marciniak, Z. S., Sehgal, S. K., Constructing Free Subgroups of Integral Group Ring Units, Proc. Amer. Math. Soc. 125 (1997), 1005-1009. MR 97f:16057
11.: Passman, D.S., Algebraic structure of group rings, Interscience, New York, 1977. MR 81d:16001
12.: Rowen, L.H., Ring theory, Academic Press, 1988. MR 89h:16001; MR 89h:16002
13.: Sehgal, S.K., Topics in group rings, Macel Dekker, 1978. MR 80j:16001
14.: Sehgal, S.K., Units in integral group rings, Longman's, Essex, 1993. MR 94m:16039

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