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GROUPS THAT INVOLVE FINITELY MANY PRIMES AND HAVE ALL SUBGROUPS SUBNORMAL II

Published online by Cambridge University Press:  30 March 2012

HOWARD SMITH
HOWARD SMITH*
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, U.S.A. e-mail: howsmith@bucknell.edu
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Abstract

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It is shown that if G is a hypercentral group with all subgroups subnormal, and if the torsion subgroup of G is a π-group for some finite set π of primes, then G is nilpotent. In the case where G is not hypercentral there is a section of G that is much like one of the well-known Heineken-Mohamed groups. It is also shown that if G is a residually nilpotent group with all subgroups subnormal whose torsion subgroup satisfies the above condition then G is nilpotent.

Keywords

20E1520F19
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 529 - 534
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Casolo, C., Torsion-free groups in which every subgroup is subnormal, Rend. Circolo Mat. Palermo. L (2001), 321–324.CrossRefGoogle Scholar
2.Casolo, C., On the structure of groups with all subgroups subnormal, J. Group Theory 5 (2002), 293–300.CrossRefGoogle Scholar
3.Hall, P., Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801.CrossRefGoogle Scholar
4.Heineken, H. and Mohamed, I. J., A group with trivial centre satisfying the normalizer condition, J. Algebra 10 (1968), 368–376.CrossRefGoogle Scholar
5.Lennox, J. C. and Stonehewer, S. E., Subnormal subgroups of groups (Clarendon, Oxford, 1987).Google Scholar
6.Möhres, W., Hyperzentrale Gruppen, deren Untergruppen alle subnormal sind, Illinois J. Math. 35 (1991), 147–157.CrossRefGoogle Scholar
7.Roseblade, J. E., On groups in which every subgroup is subnormal, J. Algebra 2 (1965), 402–412.CrossRefGoogle Scholar
8.Smith, H., Hypercentral groups with all subgroups subnormal. Bull. London Math. Soc. 15 (1983), 229–234.CrossRefGoogle Scholar
9.Smith, H., Torsion-free groups with all subgroups subnormal, Arch. Math. 76 (2001), 1–6.CrossRefGoogle Scholar
10.Smith, H., Residually nilpotent groups with all subgroups subnormal, J. Algebra. 244 (2001), 845–850.CrossRefGoogle Scholar
11.Smith, H., On non-nilpotent groups with all subgroups subnormal, Ricerche di Mat. L (2001), 217–221.Google Scholar
12.Smith, H., Groups that involve finitely many primes and have all subgroups subnormal, J. Algebra. 347 (2011), 133–142.CrossRefGoogle Scholar