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Isomorphic Subgroups of Finite p-Groups. II

Published online by Cambridge University Press:  20 November 2018

George Glauberman
George Glauberman*
Affiliation:
Department of Mathematics, The University of Chicago, Chicago, Illinois
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Suppose that we are given an isomorphism ϕ between two subgroups of index p in a finite p-group P. Let N(ϕ) be the largest subgroup of P fixed by ϕ. By a result of Sims [2, Proposition 2.1], n(ϕ) is a normal subgroup of P. In [2], we showed that P/N(ϕ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?

In this paper, we consider characteristic subgroups of P rather than normal subgroups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 1023 - 1039
Copyright
Copyright © Canadian Mathematical Society 1971

References

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3. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
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6. McLaughlin, J., Some subgroups ofSLn(F2), Illinois J. Math. 18 (1969), 108–115.Google Scholar