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Splitting over nilpotent and hypercentral residuals

Published online by Cambridge University Press:  24 October 2008

B. Hartley  and
M. J. Tomkinson
B. Hartley
Affiliation:
Mathematics Institute, University of Warwick, Coventry
M. J. Tomkinson
Affiliation:
Department of Mathematics, The University, Glasgow, W 2
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Extract

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 2 , September 1975 , pp. 215 - 226
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Carter, R. W. and Hawkes, T. O.The -normalizers of a finite soluble group. J. Algebra 5 (1967), 175–202.CrossRefGoogle Scholar
(2)Fuchs, L.Infinite Abelian groups (New York, London, Academic Press, 1973).Google Scholar
(3)Gardiner, A. D., Hartley, B. and Tomkinson, M. J.Saturated formations and Sylow structure in locally finite groups. J. Algebra 17 (1971), 177–211.CrossRefGoogle Scholar
(4)Gaschutz, W.Zur Erweiterungstheorie der endlichen Gruppen. J. reine angew. Math. 190 (1952), 93–107.CrossRefGoogle Scholar
(5)Gorenstein, D.Finite groups (Harper's Series in Modern Mathematics, New York, Harper and Row, 1968).Google Scholar
(6)Hartley, B. Groups of automorphisms of Abelian -groups (to appear in Proc. Cambridge Philos. Soc.)Google Scholar
(7)Mal'cev, A. I.On certain classes of infinite soluble groups. Mat. Sb. 28 (1951), 567–588 (Russian); Amer. Math. Soc. Transi. (2) 2 (1956), 1–21.Google Scholar
(8)Robinson, Derek J. S.Residual properties of some classes of infinite soluble groups. Proc. London Math. Soc. (3) 18 (1968), 495–520.CrossRefGoogle Scholar
(9)Robinson, Derek J. S.Finiteness conditions and generalized soluble groups, Parts 1 and 2. Ergebn. Math. Grenzgebiete 62–63 (1972).Google Scholar
(10)Robinson, D. J. S., Splitting theorems for infinite groups. Symposia Mathematica (to appear).Google Scholar
(11)Robinson, D. J. S. On the cohomology of soluble groups of finite rank. J. Pure Applied Algebra (to appear).Google Scholar
(12)Schenkman, E.The splitting of certain soluble groups. Proc. Amer. Math. Soc. 6 (1955), 286–290.CrossRefGoogle Scholar