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A lemma on finite p-groups and some consequences

Published online by Cambridge University Press:  24 October 2008

Thomas J. Laffey
Thomas J. Laffey
Affiliation:
Northern Illinois University†
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Extract

Let S be a finite p-group and let σ be a p'-automorphism of S. A well-known result of Huppert ((2), IV, (5·12), Satz) states that if σ acts trivially onΩ1(S) (ω2(S) if p = 2) then σ is the identity. In this paper, we give a short proof of this result and also give some applications of it. Our proof depends on a somewhat surprising Lemma (Lemma 1) which is of some interest in its own right.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 2 , March 1974 , pp. 133 - 137
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Blackburn, N.Automorphisms of finite p-groups. J. Algebra 3 (1966), 28–29.CrossRefGoogle Scholar
(2)Huppert, B.Endliche Gruppen Bd I (Berlin, Heidelberg, New York; Springer, 1967).CrossRefGoogle Scholar
(3)Laffey, T. J. The minimum number of generators of a finite p-group; to appear.Google Scholar
(4)Maan, A.Generators of 2-groups. Israel J. Math. 10 (1971), 158–159.CrossRefGoogle Scholar