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The Hypercentre and the n-Centre of the Unit Group of an Integral Group Ring

Published online by Cambridge University Press:  20 November 2018

Yuanlin Li
Yuanlin Li*
Affiliation:
Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, Newfoundland A1C 5S7, e-mail: yuanlin@math.mun.ca
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Abstract

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In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most 2. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n\,\ge \,2$).

Keywords

16U6020C05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 2 , 01 April 1998 , pp. 401 - 411
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Arora, Satya R., Hales, A.W. and Passi, I. B. S., Jordan decomposition and hypercentral units in integral group rings. Comm. Algebra 21(1993), 25–35.Google Scholar
2. Arora, Satya R. and Passi, I. B. S., Central height of the unit group of an integral group ring. Comm. Algebra 21(1993), 3673–3683.Google Scholar
3. Baer, R., Endlichkeitskriterien für Kommutatorgruppen. Math. Ann. 124(1952), 161–177.Google Scholar
4. Baer, R., Factorization of n-soluble and n-nilpotent groups. Proc. Amer.Math. Soc. 45(1953), 15–26.Google Scholar
5. Bovdi, A.A., Periodic normal divisors of the multiplicative group of a group ring I. Sibirsk. Mat. Zh. 9(1968), 495–498.Google Scholar
6. Baer, R., Periodic normal divisors of the multiplicative group of a group ring II. Sibirsk. Mat. Zh. 11(1970), 492–511.Google Scholar
7. Bovdi, A.A. and Sehgal, S.K., Unitary subgroup of integral group rings. Publ. Mat. 36(1992), 197–204.Google Scholar
8. Hall, M., The Theory of Groups. Macmillan, New York, 1959.Google Scholar
9. Hall, P., Verbal and marginal subgroups. J. Reine Angew. Math. 182(1940), 156–157.Google Scholar
10. Hall, P.,NilpotentGroups.Canad.Math.Congress, Univ.Alberta, Edmonton, 1957. QueenMaryCollege Math. Notes, 1970.Google Scholar
11. Hogan, G.T. and Kappe, W.P., On the Hp-problem for finite p-groups. Proc. Amer.Math. Soc. 20(1969), 450–454.Google Scholar
12. Kappe, L.C., On n-Levi groups. Arch.Math. 47(1986), 198–210.Google Scholar
13. Kappe, L.C., On power margins. J. Algebra 122(1989), 337–344.Google Scholar
14. Kappe, L.C. and Newell, M.L., On the n-centre of a group. Groups St. Andrews 2, 1989. 339–352, London Math. Soc. Lecture Note Ser. 160, Cambridge Univ. Press, Cambridge, 1991.Google Scholar
15. Parmenter, M.M., Conjugates of units in integral group rings. Comm. Algebra 23(1995), 5503–5507.Google Scholar
16. Ritter, J. and Sehgal, S.K., Integral group rings with trivial central units. Proc. Amer. Math. Soc. 108(1990), 327–329.Google Scholar
17. Sehgal, S.K., Topics in Group Rings. Marcel Dekker, New York and Basel, 1978.Google Scholar
18. Sehgal, S.K., Units in Integral Group Rings. Longman, New York, 1993.Google Scholar
19. Williamson, A., On the conjugacy classes in an integral group ring. Canad. Math. Bull. 21(1987), 491–496.Google Scholar