Hypercentral units in integral group rings

Authors:
Yuanlin Li and M. M. Parmenter

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2235-2238

MSC (2000):
Primary 16S34, 20C07

Published electronically:
January 23, 2001

MathSciNet review:
1823905

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this note, we show that when is a torsion group the second center of the group of units of the integral group ring is generated by its torsion subgroup and by the center of . This extends a result of Arora and Passi (1993) from finite groups to torsion groups, and completes the characterization of hypercentral units in when is a torsion group.

**[1]**Satya R. Arora, A. W. Hales, and I. B. S. Passi,*Jordan decomposition and hypercentral units in integral group rings*, Comm. Algebra**21**(1993), no. 1, 25–35. MR**1194548**, 10.1080/00927879208824548**[2]**Satya R. Arora and I. B. S. Passi,*Central height of the unit group of an integral group ring*, Comm. Algebra**21**(1993), no. 10, 3673–3683. MR**1231624**, 10.1080/00927879308824756**[3]**Norman Blackburn,*Finite groups in which the nonnormal subgroups have nontrivial intersection*, J. Algebra**3**(1966), 30–37. MR**0190229****[4]**A. A. Bovdi,*Periodic normal subgroups of the multiplicative group of a group ring. II*, Sibirsk. Mat. Z.**11**(1970), 492–511 (Russian). MR**0279207****[5]**Yuanlin Li,*The hypercentre and the 𝑛-centre of the unit group of an integral group ring*, Canad. J. Math.**50**(1998), no. 2, 401–411. MR**1618302**, 10.4153/CJM-1998-021-2**[6]**Yuanlin Li, M.M. Parmenter and S.K. Sehgal, On the normalizer property for integral group rings, Commun. in Alg. 27 (1999), 4217-4223. CMP**99:17****[7]**M. M. Parmenter,*Conjugates of units in integral group rings*, Comm. Algebra**23**(1995), no. 14, 5503–5507. MR**1363619**, 10.1080/00927879508825548**[8]**M. M. Parmenter,*Central units in integral group rings*, Algebra, Trends Math., Birkhäuser, Basel, 1999, pp. 111–116. MR**1690793****[9]**Sudarshan K. Sehgal,*Topics in group rings*, Monographs and Textbooks in Pure and Applied Math., vol. 50, Marcel Dekker, Inc., New York, 1978. MR**508515****[10]**S. K. Sehgal,*Units in integral group rings*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR**1242557**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16S34,
20C07

Retrieve articles in all journals with MSC (2000): 16S34, 20C07

Additional Information

**Yuanlin Li**

Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, New Foundland, Canada A1C 5S7

**M. M. Parmenter**

Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, New Foundland, Canada A1C 5S7

DOI:
https://doi.org/10.1090/S0002-9939-01-05848-8

Received by editor(s):
August 3, 1999

Received by editor(s) in revised form:
December 24, 1999

Published electronically:
January 23, 2001

Additional Notes:
This research was supported in part by grants from the Natural Sciences and Engineering Research Council.

Communicated by:
Steven D. Smith

Article copyright:
© Copyright 2001
American Mathematical Society