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On the order of the Sylow subgroups of the automorphism group of a finite group

Published online by Cambridge University Press:  18 May 2009

K. H. Hyde
K. H. Hyde
Affiliation:
Weber State College, Ogden, Utah, U.S.A.
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Given any finite group G, we wish to determine a relationship between the highest power of a prime p dividing the order of G, denoted by |G|p, and |A(G)|p, where A(G) is the automorphism group of G. It was shown by Herstein and Adney [8] that |A(G)|p ≧ p whenever |G|p= ≧P2. Later Scott [16] showed that A(G)p≧P2. For the special case where G is abelian, Hilton [9] proved that Adney [1] showed that this result holds if a Sylow p-subgroup of G is abelian, and gave an example where |G|p= p4 and |A(G)|P =p2. We are able to show in Theorem 4.5 that, if |G|p = ≧ p5, then |A(G)| = ≧ p3.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 11 , Issue 2 , July 1970 , pp. 88 - 96
Copyright
Copyright © Glasgow Mathematical Journal Trust 1970

References

REFERENCES

1.Adney, J. E., On the power of a prime dividing the order of a group of automorphisms, Proc. Amer. Math. Soc. 8 (1957), 627–633.CrossRefGoogle Scholar
2.Adney, J. E. and Yen, T., Automorphisms of a p-group, Illinois J. Math. 9 (1965), 137–143.CrossRefGoogle Scholar
3.Faudree, R., A note on the automorphism group of a p-group, Proc. Amer. Math. Soc. 19 (1968), 1379–1382.Google Scholar
4.Fitting, H., Die Gruppe der zentralen Automorphismen einer Gruppe mit Hauptreihe, Math. Ann. 114 (1937), 355–372.Google Scholar
5.Gaschütz, W., Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen, Journal of Algebra 4 (1966), 1–2.Google Scholar
6.Green, J. A., On the number of automorphisms of a finite group, Proc. Roy. Soc. (A) 237 (1956), 574–581.Google Scholar
7.Hall, M., The theory of groups (New York, 1959).Google Scholar
8.Herstein, I. N. and Adney, J. E., A note on the automorphism group of a finite group, Amer. Math. Monthly 59 (1952), 309–310.Google Scholar
9.Hilton, H., On the order of the group of automorphisms of an abelian group, Messenger of Mathematics II 38 (1909), 132–134.Google Scholar
10.Howarth, J. C., On the power of a prime dividing the order of the automorphism group of a finite group, Proc. Glasgow Math. Assoc. 4 (1960), 163–170.Google Scholar
11.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
12.Otto, A. D., Central automorphisms of a finite p-group, Trans. Amer. Math. Soc. 125 (1966), 280–287.Google Scholar
13.Ranum, A., The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group, Trans. Amer. Math. Soc. 8 (1907), 71–91.CrossRefGoogle Scholar
14.Schenkman, E., The existence of outer automorphisms of some nilpotent groups of class 2, Proc. Amer. Math. Soc. 6 (1955), 6–11.Google Scholar
15.Schur, I., Über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20–50.Google Scholar
16.Scott, W. R., On the order of the automorphism group of a finite group, Proc. Amer. Math. Soc. 5 (1954), 23–24.CrossRefGoogle Scholar
17.Wiegold, J., Multiplicators and groups with finite central factor-groups, Math. Zeit. 89 (1965), 345–347.CrossRefGoogle Scholar