On the normalizer property for integral group rings

Li, Yuanlin; Sehgal, Sudarshan K.; Parmenter, M. M.

doi:10.1080/00927879908826692

Cited by 41 publications

(18 citation statements)

References 3 publications

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“…and there exists a positive integer n = n(u) such that u n hu −n = h. It follows from Theorem 1.2 of [7] that the exponent of Z 1 (G) is 2, and so for all u 2 ∈ Z 2 (V (ZG)) and all g ∈ G, we have [u We remark that the relationship Z 2 (V (ZG)) ⊆ N V (ZG) (G), obtained in the above proof, was also noted in [6] and, as a consequence, some partial results on the problem of when Z 2 (V (ZG)) = T · Z 1 (V (ZG)) were obtained in that paper.…”

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confidence: 82%

Hypercentral units in integral group rings

Parmenter

2001

Proc. Amer. Math. Soc.

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Abstract. In this note, we show that when G is a torsion group the second center of the group of units U (ZG) of the integral group ring ZG is generated by its torsion subgroup and by the center of U (ZG). This extends a result of from finite groups to torsion groups, and completes the characterization of hypercentral units in ZG when G is a torsion group.Let ZG denote the integral group ring of a torsion group G, U (ZG) the group of units of such a group ring and V (ZG) the subgroup of units of augmentation 1.In the case where G is finite, the ascending central serieswhere T denotes the torsion subgroup of Z 2 (V (ZG)). The first of these results was extended to torsion groups in [5], and the question of whether or not the second result could be similary extended was Open Problem 5 in [8].In this note, we show that the second result can indeed be extended to torsion groups. Although Blackburn's theorem [3] on the intersection of nonnormal subgroups of finite groups played a significant role in the investigation of the structure of the second center in [2], this result cannot be applied directly to our case. Instead, our argument focusses on the importance of Bass cyclic, bicyclic and Hoechsmann units in integral group rings.Notation and terminology will follow that in [10]. The following lemma is crucial to our approach. Although the argument given can be found in [5] (as part of the proof of Theorem 2), we include it here for completeness.Since cg is of finite order, it follows that c is also of finite order and hence is in G [9, p.462 is not contained in Z 1 (V (ZG)), then there must exist a group element h in Z 2 (V (ZG))\Z 1 (V (ZG)). But then if u is any unit in ZG, [u, h] = h 0 ∈ Z 1 (G) and there exists a positive integer n = n(u) such that u n hu −n = h. It follows from Theorem 1.2 of [7] that the exponent of Z 1 (G) is 2, and so for all u 2 ∈ Z 2 (V (ZG))

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confidence: 82%

Hypercentral units in integral group rings

Parmenter

2001

Proc. Amer. Math. Soc.

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“…Petit Lobao and Polcino Milies [9]have confirmed (Nor) for Frobenius groups. There are other groups for which (Nor) holds (see Li, Parmenter and Sehgal [6]). In this note, we study complete monomial groups.…”

Section: Introductionmentioning

confidence: 99%

The Normalizer Property for Integral Group Rings of Complete Monomial Groups

Lobão

Sehgal²

2003

Communications in Algebra

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“…Recently, Li, Sehgal and Parmenter [6] proved that the normalizer property holds for any finite group G with RðGÞ ¼ 1, where RðGÞ denotes the intersection of all nonnormal subgroups of G. Marciniak and Roggenkamp [7] showed that the normalizer property holds for finite metabelian groups having an abelian Sylow 2-subgroup. Petit Lobão and Polcino Milies [12] proved that the normalizer property holds for finite Frobenius groups.…”

Section: Introductionmentioning

confidence: 99%