202 research outputs found

    Frobenius groups of automorphisms and their fixed points

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    Suppose that a finite group GG admits a Frobenius group of automorphisms FHFH with kernel FF and complement HH such that the fixed-point subgroup of FF is trivial: CG(F)=1C_G(F)=1. In this situation various properties of GG are shown to be close to the corresponding properties of CG(H)C_G(H). By using Clifford's theorem it is proved that the order G|G| is bounded in terms of H|H| and CG(H)|C_G(H)|, the rank of GG is bounded in terms of H|H| and the rank of CG(H)C_G(H), and that GG is nilpotent if CG(H)C_G(H) is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of GG in the case of metacyclic FHFH. The exponent of GG is bounded in terms of FH|FH| and the exponent of CG(H)C_G(H) by using Lazard's Lie algebra associated with the Jennings--Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of GG is bounded in terms of H|H| and the nilpotency class of CG(H)C_G(H) by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.Comment: 31 page

    The strong Anick conjecture is true

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    Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K. We also find a large new class of wild automorphisms of K which is not covered by the results of Umirbaev. Finally, we study the lifting problem for automorphisms and coordinates of polynomial algebras, free metabelian algebras and free associative algebras and obtain some interesting new results.Comment: 25 pages, corrected typos and acknowledgement

    Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

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    We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates
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