202 research outputs found
Frobenius groups of automorphisms and their fixed points
Suppose that a finite group admits a Frobenius group of automorphisms
with kernel and complement such that the fixed-point subgroup of
is trivial: . In this situation various properties of are
shown to be close to the corresponding properties of . By using
Clifford's theorem it is proved that the order is bounded in terms of
and , the rank of is bounded in terms of and the rank
of , and that is nilpotent if is nilpotent. Lie ring
methods are used for bounding the exponent and the nilpotency class of in
the case of metacyclic . The exponent of is bounded in terms of
and the exponent of by using Lazard's Lie algebra associated with the
Jennings--Zassenhaus filtration and its connection with powerful subgroups. The
nilpotency class of is bounded in terms of and the nilpotency class
of 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
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
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
- …