Una nota sui gruppi policiclici che ammettono un automorfismo con numero di Reidemeister finito

LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g ⁻¹ yg ^θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ. In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2 ⁿ⁻¹ .

     

On periodic solvable groups having automorphisms with nilpotent fixed point groups

Letp be a prime,G a periodic solvablep′-group acted on by an elementary groupV of orderp ². We show that ifC _G(v) is abelian for eachv ∈V ^# thenG has nilpotent derived group, and ifp=2 andC _G(v) is nilpotent for eachv ∈V ^# thenG is metanilpotent. Earlier results of this kind were known only for finite groups.     

Automorfismi spezzanti di ordine primo

LetG be a group andα εAut(G);α is calledn-splitting if gg^α...g^{α n-1}=1 ∀g εG. In this note we study the structure of finite groups admitting an-splitting automorphism of orderp (p an odd prime number).

     

Polynomial mappings of groups

A mapping ϕ of a groupG to a groupF is said to be polynomial if it trivializes after several consecutive applications of operatorsD _h ,h ∈G, defined byD _h ϕ(g)=ϕ(g) ⁻¹ ϕ(gh). We study polynomial mappings of groups, mainly to nilpotent groups. In particular, we prove that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any polynomial mappingG→F to a nilpotent groupF splits into a homomorphismG→G’ to a nilpotent groupG’ and a polynomial mappingG’→F. We apply the obtained results to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group. This work was supported by NSF, Grants DMS-9706057 and 0070566.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

UNA generalizzazione del problema di burnside relativo all'esponente 4

An automorphism σ of order (a divisor of)n of the groupG is calledn-splitting if for everyg∈G. In this paper we prove that a 2-group admitting a 4-splitting automorphism, is locally finite.     

On the centralizer and the commutator subgroup of an automorphism

Let φ be an automorphism of a group G. In this paper, we study the influence of its centralizer on its commutator subgroup when G is polycyclic or metabelian. For instance, when G is metabelian and φ fixed-point-free of prime order p, we prove that is nilpotent of class ≤ p. Also, when G is polycyclic and φ of order 2, we show that if is finite, then so are and .     

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X ²|<|X|², whereX ²={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism. Dedicated to John G. Thompson, the Wolf Prize Laureate in Mathematics for 1992 The first author wishes to thank the Department of Mathematics in the University of Napoli for their hospitality during the preparation of this paper.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.     

Gruppi che ammettono un automorfismo 4-spezzante

LetG be a group admitting a 4-splitting automorphism (i.e. an automorphism σ such that for everyg∈G). In this paper we prove that ifG≠1 is solvable with derived lengthd thenG′ is nilpotent of class not greater than (4 ^d−1−1)/3.     

Products of groups and group classes

Letχ be a Schunck class, and let the finite groupG=AB=BC=AC be the product of two nilpotent subgroupsA andB andχ-subgroupC. If for every common prime divisorp of the orders ofA andB the cyclic group of orderp is anχ-group, thenG is anχ-group. This generalizes earlier results of O. Kegel and F. Peterson. Some related results for groups of the formG=AB=AK=BK, whereK is a nilpotent normal subgroup ofG andA andB areχ-groups for some saturated formationχ, are also proved.     

Automorfismi spezzanti di ordine 2

LetG be a group and α an automorphism ofG; α is calledn-splitting if for allg∈G. In this note we study the structure of finite groups admitting an-splitting automorphism of order 2.

     

Almost fixed-point-free automorphisms of prime order

Let ϕ be an automorphism of prime order p of the group G with C _G (ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.     

Elementary abelian operator groups

Letp be a prime,n a positive integer. Suppose thatG is a finite solvablep'-group acted on by an elementary abelianp-groupA. We prove that ifC _G (ϕ) is of nilpotent length at mostn for every nontrivial element ϕ ofA and |A|≥p ⁿ⁺¹ thenG is of nilpotent length at mostn+1.     

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.