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An example in the theory of solubl groups

Published online by Cambridge University Press:  24 October 2008

T. O. Hawkes
T. O. Hawkes
Affiliation:
University of Warwick, Coventry
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Extract

Let G be a finite soluble group. In (1) Alperin proves that two system normalizers of G contained in the same Carter subgroup C of G are conjugate in C. In recent unpublished work G.A.Chambers of the University of Wisconsin has proved that, if is a saturated formation, the -normalizers of an A-group are pronormal subgruops; hence, in particular, that two -normalizers contained in an -projector E of an A-group are conjugate in E. In this note we describe an example which shows that in Alperin's theorem the class of nilpotent groups cannot in general be replaced by an arbitary saturated formation without some restriction on the class of soluble groups under consideration. we prove

PROPOSITION. There exists a saturated formationand a group G which has two-normalizers E1and E2contained in an-projector F of G such that E1and E2are not conjugate in F.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 1 , January 1970 , pp. 13 - 16
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Alperin, J. L.System normalizers and Carter subgroups. J. Algebra 1 (1964), 355–366.CrossRefGoogle Scholar
(2)Carter, R. W. and Hawkes, T. O.The -norma1izers of a finite soluble group. J. Algebra 5 (1967), 175–202.CrossRefGoogle Scholar
(3)Fischer, B., Gaschütz, W. and Hartley, B.Injektoren endlicher auflösbarer Gruppen Math. Z. 102 (1967), 337–339.CrossRefGoogle Scholar
(4)Hall, P.On the system normalizers of a soluble group. Proc. London Math. Soc. 43 (1937), 507–528.Google Scholar