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Locally defined Fitting classes

Published online by Cambridge University Press:  09 April 2009

Patrick D' Arcy
Affiliation:
University of Auckland, Auckland, New Zealand
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Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1975

References

Fischer, B., Habilitationsschrift (Universitat Frankfurt.)Google Scholar
Fischer, B., Gaschütz, W., and Hartley, B. (1967), ‘Injektoren endlicher auflösbarer Gruppen’, Math. Z. 102, 337339.CrossRefGoogle Scholar
Graddon, C. (1971), ‘F-reducers in finite soluble groups’, J. Algebra 18, 574587.CrossRefGoogle Scholar
Hartley, B. (1969), ‘On Fischer's dualization of formation theory, Proc. London. Math. Soc. (3) 19, 193207.CrossRefGoogle Scholar