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Isomorphic Subgroups of Finite p-Groups. I

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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Let p be a prime and P be a p-subgroup of a finite group G. Suppose that g ∈ G and that P ∩ Pg has index p in P. In [4], we assumed that g normalizes no non-identity normal subgroup of P. We obtained some bounds on the order of P and some applications to the case in which p = 2 and P is a Sylow 2-subgroup of 〈P, Pg〉. In this paper, we examine this situation further by considering the isomorphism ϕ of P ∩ Pg-l onto P ∩ Pg given by ϕ(x) = xg. We actually consider arbitrary isomorphisms ϕ between two subgroups of index p in P. However, an easy argument (Lemma 2.3) shows that every such ϕ can be obtained as above for some G and some g. We obtain some results concerning the nilpotence class rather than the order of P.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 983 - 1022
Copyright
Copyright © Canadian Mathematical Society 1971

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