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The ranks of central factor and commutator groups

Published online by Cambridge University Press:  14 August 2012

LEONID A. KURDACHENKO  and
PAVEL SHUMYATSKY
LEONID A. KURDACHENKO
Affiliation:
Department of Algebra, National University of Dnepropetrovsk 72 Gagarin Av., Dnepropetrovsk, Ukraine49010. e-mail: [email protected]
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil. e-mail: [email protected]
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Abstract

The Schur Theorem says that if G is a group whose center Z(G) has finite index n, then the order of the derived group G′ is finite and bounded by a number depending only on n. In this paper we show that if G is a finite group such that G/Z(G) has rank r, then the rank of G′ is r-bounded. We also show that a similar result holds for a large class of infinite groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 1 , January 2013 , pp. 63 - 69
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Adian, S. I.On some torsion-free groups. Math. USSR-Izv. 5 (1971), 475–484.Google Scholar
[2]Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D.Analytic Pro-p Groups (Cambridge University Press, 1991).Google Scholar
[3]Dixon, M. R., Kurdachenko, L. A. and Polyakov, N. V.On some ranks of infinite groups. Ricerche Mat. 56 (2007), 43–59.CrossRefGoogle Scholar
[4]Franciosi, S., Giovanni, F. de and Kurdachenko, L. A.The Schur property and groups with uniform conjugacy classes. J. Algebra 174 (1995), 823–847.CrossRefGoogle Scholar
[5]Gorenstein, D.Finite Simple Groups: An Introduction to Their Classification (Plenum Press, 1982).CrossRefGoogle Scholar
[6]Kegel, O. H. and Wehrfritz, B. F. A.Locally Finite Groups (North-Holland, 1973).Google Scholar
[7]Kurdachenko, L. A. On groups with minimax conjugacy classes. In: Infinite groups and adjoining algebraic structures. Naukova Dumka Kiev (1993), 160–177.Google Scholar
[8]Mann, A.The exponents of central factor and commutator groups. J. Group Theory (4)10 (2007), 435–436.CrossRefGoogle Scholar
[9]Olshanskii, A. Yu.Geometry of Defining Relations in Groups (Kluwer Academic Publishers, 1991).CrossRefGoogle Scholar
[10]Polovicky, Ya. D.Groups with extremal classes of conjugate elements. Sibir. Math. J. 5 (1964), 891–895.Google Scholar
[11]Robinson, D. J. S.Finiteness Conditions and Generalized Soluble Groups, Part 1 (Springer Verlag, 1972).CrossRefGoogle Scholar
[12]Shalev, A.Characterization of p-Adic analytic groups in terms of Wreath products. J. Algebra 145 (1992), 204–208.CrossRefGoogle Scholar
[13]Zassenhaus, H.Beweis eines satzes über diskrete gruppen. Abh. Math. Sem. Univ. Hamburg 12 (1938), 289–312.CrossRefGoogle Scholar
[14]Zelmanov, E.The solution of the Restricted Burnside Problem for groups of odd exponent. Math. USSR Izv. 36 (1991), 41–60.CrossRefGoogle Scholar
[15]Zelmanov, E.The solution of the Restricted Burnside Problem for 2-groups. Math. Sb. 182 (1991), 568–592.Google Scholar