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On the power of a prime dividing the order of the automorphism group of a finite group

Published online by Cambridge University Press:  18 May 2009

J. C. Howarth
J. C. Howarth
Affiliation:
The University Glasgow
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The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequality

Later Green [2] improved this estimate to

In the Present paper this will be revised, for sufficiently large h, to

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 4 , September 1960 , pp. 163 - 170
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

1.Eilenberg, S. and MacLane, S., Cohomology theory in abstract groups I, Ann. of Math. 48 (1947), 51.CrossRefGoogle Scholar
2.Green, J. A., On the number of automorphisms of a finite group, Proc. Roy. Soc. A 237 (1956), 574–581.Google Scholar
3.Hughes, N. J. S., The structure and order of the group of central automorphisms of a finite group, Proc. London Math. Soc.(2) 52 (1951), 377–385.Google Scholar
4.Ledermann, W. and Neumann, B. H., On the order of the automorphism group of a finite group II, Proc. Roy. Soc. A 235 (1956), 235–246.Google Scholar
5.Ranum, A., The group of classes of congruent matrices and group of isomorphisms of any Abelian group, Trans. Amer. Math. Soc. 8 (1907), 71–91.CrossRefGoogle Scholar
6.Schur, J., Über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20–50.Google Scholar
7.Shoda, K., Über die Automorphismen einer endlichen Abelschen Gruppe, Math. Annalen 100 (1928), 674–686.CrossRefGoogle Scholar