p-groups of automorphisms of Abelian p-groups

Khukhro, E. I. (2000) p-groups of automorphisms of Abelian p-groups. Algebra and Logic, 39 (3). pp. 207-214. ISSN 0002-5232

Full content URL: http://dx.doi.org/10.1007/BF02681764


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We consider the action of a p-group G on an Abelian p-group A, with the latter treated as a faithful right �,G-module. Our aim is to establish a connection between exponents of the kernels under the induced action of G on elementary p-group A-pA and Ω1 and (A) = x � A px = 0; the kernels are denoted by CG;(A/pA) and CG(Ω1(A)). respectively. It is proved that if the exponent of one of the kernels CG,(A/pA) or CG(Ω1(A)) is finite then the other also has a finite exponent bounded in terms of the first; moreover, these kernels are nilpotent. In one case we impose the � additional restriction � piA= 0. And the wreath product Cp� G of a quasicyclic group and i=1 an arbitrary p-group G shows that this condition cannot be dropped. The results obtained are used to confirm, for one particular case, the conjecture on the boundedness of a derived length of a finite group with an automorphism of order 2 all of whose fixed points are central. (The solubility of such groups, and also the reduction to the case of 2-groups, were established in 1.) © 2000 Plenum Publishing Corporation.

Keywords:Algebra, Logic
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:15741
Deposited On:19 Nov 2014 14:25

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