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Local definitions of local homomorphs and formations of finite groups

Published online by Cambridge University Press:  17 April 2009

P. Förster  and
E. Salomon
P. Förster
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia;
E. Salomon
Affiliation:
Fachbereich Mathematik, J. Gutenberg-Universität, 6500 Mainz, Federal Republic of Germany.
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Abstract

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It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 1 , February 1985 , pp. 5 - 34
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Barnes, D.W. and Kegel, O.H., “Gaschütz functors on finite soluble groups”, Math. Z. 94 (1966), 134–142.CrossRefGoogle Scholar
[2]Bryant, R.M., Bryce, R.A. and Hartley, B., “The formation generated by a finite group”, Bull. Austral. Math. Soc. 2 (1970), 347–357.CrossRefGoogle Scholar
[3]Carter, R.W. and Hawkes, T.O., “The of a finite soluble group”, J. Algebra 5 (1967), 175–202.CrossRefGoogle Scholar
[4]Doerk, K., “Die maximale lokale Erklärung einer gesättigten Formation”, Math. Z. 133 (1973), 133–135.CrossRefGoogle Scholar
[5]Doerk, K., “Über Homomorphe endlicher auflösbarer Gruppen”, J. Algebra 30 (1974), 12–30.CrossRefGoogle Scholar
[6]Förster, P., “Closure operations for Schunek classes and formations of finite solvable groups”, Math. Proc. Cambridge Philos. Soc. 85 (1979), 253–259.CrossRefGoogle Scholar
[7]Förster, P., “Projektive Klassen endlicher Gruppen. I Schunck- und Gaschützklassen”, Math. Z. 186 (1984), 149–178.CrossRefGoogle Scholar
[8]Förster, P., “projektive Klassen endlicher Gruppen. II Gesättigte Formationen”, publ. Sec. Mat. Univ. Aut. Barcelona, Arch. Math. (to appear).Google Scholar
[9]Förster, P., “Pull-backs of projectors in finite groups”, Proc. Amer. Math. Soc. (to appear).Google Scholar
[10]Šemetkov, L.A., Formations of finite groups (Russian) (Nauka, Moscow, 1978).Google Scholar