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ON SOLUBILITY OF GROUPS WITH BOUNDED CENTRALIZER CHAINS

Published online by Cambridge University Press:  01 January 2009

E. I. KHUKHRO
E. I. KHUKHRO*
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk 630090, Russia e-mail: khukhro@yahoo.co.uk
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Abstract

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The c-dimension of a group is the maximum length of a chain of nested centralizers. It is proved that a periodic locally soluble group of finite c-dimension k is soluble of derived length bounded in terms of k, and the rank of its quotient by the Hirsch–Plotkin radical is bounded in terms of k. Corollary: a pseudo-(finite soluble) group of finite c-dimension k is soluble of derived length bounded in terms of k.

Keywords

Primary: 20D10Secondary: 03C2020D4520F1620F22
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 1 , January 2009 , pp. 49 - 54
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Berger, T. R. and Gross, F., 2-length and the derived length of a Sylow 2-subgroup, Proc. Lond. Math. Soc. 34 (3) (1977), 520–534.CrossRefGoogle Scholar
2.Bludov, V. V., On locally nilpotent groups with the minimality condition for centralizers, Algebra Logika 37 (1998), 270–278; English transl. in Algebra and Logic 37 (1998), 151–156.CrossRefGoogle Scholar
3.Bryant, R. M., Groups with minimal condition on centralizers, J. Algebra 60 (1979), 371–383.CrossRefGoogle Scholar
4.Bryant, R. M. and Hartley, B., Periodic locally soluble groups with the minimal condition on centralizers, J. Algebra 61 (1979), 328–334.CrossRefGoogle Scholar
5.Bryukhanova, E. G., Connection between the 2-length and the derived length of a Sylow 2-subgroup of a finite solvable group, Mat. Zametki 29 (1981), 161–170; English transl. in Math. Notes 29 (1981), 85–90.Google Scholar
6.Derakhshan, J. and Wagner, F. O., Nilpotency in groups with chain conditions, Quart. J. Math. Oxford 48 (1997), 453–468.CrossRefGoogle Scholar
7.Duncan, A. J., Kazachkov, I. V. and Remeslennikov, V. N., Centraliser dimension and universal classes of groups, Siberian Electronic Math. Rep. 3 (2006), 197–215; http://semr.math.nsc.ru.Google Scholar
8.Hall, P. and Higman, G., On the p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. Lond. Math. Soc. 6 (3) (1956), 1–42.CrossRefGoogle Scholar
9.Kegel, O., Four lectures on Sylow theory in locally finite groups, in Group Theory Proc. Int. Conf. Singapore, 1987, Walter de Gruyter, Amsterdam, 1989, pp. 3–27.CrossRefGoogle Scholar
10.Kovács, L. G., On finite soluble groups, Math. Z. 103 (1968), 37–39.CrossRefGoogle Scholar
11.Macpherson, D. and Tent, K., Stable pseudofinite groups, J. Algebra 312 (2007), 550–561.CrossRefGoogle Scholar
12.Myasnikov, A. and Shumyatsky, P., Discriminating groups and c-dimension, J. Group Theory 7 (2004), 135–142.Google Scholar
13.Thompson, J., Automorphisms of solvable groups, J. Algebra 1 (1964), 259–267.CrossRefGoogle Scholar
14.Wagner, F. O., Stable groups, mostly of finite exponent, Notre Dame J. Formal Logic 34 (1993) 183–192.CrossRefGoogle Scholar
15.Wagner, F. O., Nilpotency in groups with the minimal condition on centralizers, J. Algebra 217 (1999), 448–460.CrossRefGoogle Scholar
16.Wilson, J. S., On pseudofinite simple groups, J. Lond. Math. Soc. 51 (2) (1995), 471–490.CrossRefGoogle Scholar