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Autoclinisms and automorphisms of finite groups

Published online by Cambridge University Press:  17 April 2009

Ratje Reimers  and
Jürgen Tappe
Ratje Reimers
Affiliation:
Institut für Reine und Angewandte Mathematik, Rheinisch-Westfäliscne Technische Hochschule, Aachen, Germany;
Jürgen Tappe
Affiliation:
Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany.
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Abstract

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Let Γ be a family of isoclinic finite groups, and let acl(Γ) be the group of autoclinisms of Γ. In this paper we prove the following formula:

where the sum is taken over a complete system of stem groups S in Γ. This result is due to P. Hall, who outlined a proof in his paper, “On groups of automorphisms” (J. reine angew. Math). 182 (1940), 194–204, using presentations, whereas in this paper we consider stem groups in terms of central extensions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 3 , December 1975 , pp. 439 - 449
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Fuchs, L., Abelian groups, 3rd ed. (Pergamon Press, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1960; reprinted 1967).Google Scholar
[2]Hall, P., “The classification of prime power groups”, J. reine angew. Math. 182 (1940), 130–141.CrossRefGoogle Scholar
[3]Hall, P., “On groups of automorphisms”, J. reine angew. Math. 182 (1940), 191–204.Google Scholar
[4]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[5]Reimers, Ratje, “Die Berechnung der Stammgruppen einer Isoklinismen-familie”,(Diplomarbeit, Aachen, 1972).Google Scholar
[6]Sanders, P.R., “The central automorphisms of a finite group”, J. London Math. Soc. 44 (1969), 225–228.CrossRefGoogle Scholar