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On Fixed-Point-Free Automorphisms

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If you have a question about this talk, please contact David Craven.

If a finite group A acts on a finite group G in such a way that CG(A)=1, then one can often say something about the structure of G given properties of A. This idea has long been a source of strong and useful results within finite group theory. I will begin with a brief historical overview of results of this kind before talking in more detail about Khukhro’s recent work in this area, and how a question which I have considered is related to his work. Namely, if we have a group RF where R is cyclic of prime order and F is nilpotent with F=[F,R], which acts on a group G such that CG(F)=1, then is F(CG(R)) ≤ F(G)? I will finish by outlining directions for further work and potential obstacles that one might encounter in pursuing these.

This talk is part of the Algebra seminar series.

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