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Normal -Fitting classes

  • J. C. Beidleman (a1) and M. J. Tomkinson (a2)

Abstract

The authors together with M. J. Karbe [Ill. J. Math. 33 (1989) 333–359] have considered Fitting classes of -groups and, under some rather strong restrictions, obtained an existence and conjugacy theorem for -injectors. Results of Menegazzo and Newell show that these restrictions are, in fact, necessary.

The Fitting class is normal if, for each is the unique -injector of G. is abelian normal if, for each. For finite soluble groups these two concepts coincide but the class of Černikov-by-nilpotent -groups is an example of a nonabelian normal Fitting class of -groups. In all known examples in which -injectors exist is closely associated with some normal Fitting class (the Černikov-by-nilpotent groups arise from studying the locally nilpotent injectors).

Here we investigate normal Fitting classes further, paying particular attention to the distinctions between abelian and nonabelian normal Fitting classes. Products and intersections with (abelian) normal Fitting classes lead to further examples of Fitting classes satisfying the conditions of the existence and conjugacy theorem.

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Copyright

References

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1.Beidleman, J. C. and Brewster, B., Strict normality in Fitting classes I, J. Algebra 51 (1978), 211217.
2.Beidleman, J. C., Karbe, M. J. and Tomkinson, M. J., Fitting classes of -groups I, Illinois J. Math. 33 (1989), 333359.
3.Beidleman, J. C. and Tomkinson, M. J., Fitting classes of -groups, II: Lockett's *-construction, Ricerche Mat. 37 (1988), 283297.
4.Beidleman, J. C. and Tomkinson, M. J., Hypercentral and nilpotent injectors of -groups, Comm. Algebra 18 (1990), 43074321.
5.Blessenohl, D. and Gaschütz, W., Über normale Schunck- und Fittingklassen, Math. Z. 118 (1970), 18.
6.Lausch, H., On normal Fitting classes, Math. Z. 130 (1973), 6772.
7.Menegazzo, F. and Newell, M. L., Injectors and Fitting classes of -groups, Atti Acad. Naz. Lined Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), 629637.

Normal -Fitting classes

  • J. C. Beidleman (a1) and M. J. Tomkinson (a2)

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