Continuous families of rational surface automorphisms with positive entropy

For any k we construct k-parameter families of rational surface automorphisms with positive entropy. These are automorphisms of surfaces ${\mathcal{X}}$ , which are constructed from iterated blowups over the projective plane. In certain cases we are able to determine the exact automorphism group of ${\mathcal{X}}$ , as well as when two of these surfaces are inequivalent.     

Crossed products and entropy of automorphisms

Let be an exact C∗-algebra, let G be a locally compact group, and let be a C∗-dynamical system. Each automorphism α_g induces a spatial automorphism Ad_λg on the reduced crossed product . In this paper we examine the question, first raised by E. Størmer, of when the topological entropies of α_g and Ad_λg coincide. This had been answered by N. Brown for the particular case of discrete abelian groups. Using different methods, we extend his result to preservation of entropy for α_g when the subgroup of Aut(G) generated by the corresponding inner automorphism Ad_g has compact closure. This property is satisfied by all elements of a wide class of groups called locally [FIA]⁻. This class includes all abelian groups, both discrete and continuous, as well as all compact groups.     

Coleman automorphisms of finite groups

An automorphism of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer of G in the units of the integral group . Let Out be the image of these automorphisms in Out. We prove that Out is always an abelian group (based on previous work of E. C. Dade, who showed that Out is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out is a -group. If O, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of has order 2, then , where is the center of . This improves an earlier result of S. Jackowski and Z. Marciniak. Received: 26 May 2000; in final form: 5 October 2000 / Published online: 19 October 2001     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and [C_G(u), C_G(v),...,C_G(v)]=1{[C_G(u), C_G(v),dots,C_G(v)]=1} for any u,v ? B{1}{u,vin B{setminus}{1}}, where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every b ? B{1}{b in B{setminus}{1}}, then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

Length-type parameters of finite groups with almost unipotent automorphisms

Let α be an automorphism of a finite group G. For a positive integer n, let E _G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E _G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E _G,n (α). For soluble G we prove that if, for some n, the Fitting height of E _G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F _h* (H) = H, where F ₀* (H) = 1, and F _i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F _i *(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E _G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE _G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Groups of automorphisms of finite p-groups

Thompson [1] showed that if p is an odd prime number, A is a p-group of operators of the finite group P in which the Frattini subgroup (P) is elementary and central, and P/(P) is a free Z_pA-module, then Cp(A) covers C_P/(P)(A). Then he proposed the question of whether it is possible in this theorem to weaken the hypothesis that (P) be elementary and central. In the work it is shown that this hypothesis may be replaced by a much weaker one; it is sufficient that P be met-Abelian and have nilpotence class 相似文献   

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

On finite groups with automorphisms whose fixed points are Engel

The main result of the paper is the following theorem. Let q be a prime, n a positive integer, and A an elementary abelian group of order q². Suppose that A acts coprimely on a finite group G and assume that for each \({a \in A^{\#}}\) every element of C_G(a) is n-Engel in G. Then the group G is k-Engel for some \({\{n,q\}}\)-bounded number k.     

Bundle automorphisms for fiberG-structures of finite type

In this paper we study geometric settings where a Lie group preserving a measurable field of measurable Riemannian metrics on the fibers of a smooth fiber bundle must actually preserve a measurable field of smooth Riemannian metrics. For ergodic actions on bundles with compact fiber this will imply that the standard fiber is a homogeneous space for a compact Lie group. In particular we show this conclusion holds for a semisimple Lie group of higher real rank (or a lattice subgroup) preserving a finite measure and either a field of connections or pseudo-Riemannian metrics when the fiber is compact and of low dimension.Research completed while a member of the University of Chicago Mathematics Department.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.