Finite central height in linear groups

J.D. Dixon has characterized those pairs (n,F), where n is a positive integer and F a field, for which every locally nilpotent subgroup of GL(n,F) is nilpotent. He showed further that these pairs (n,F) have the stronger property that there is a bound on the nilpotency class of the nilpotent subgroups of GL(n,F). In this note we show that these pairs (n,F) have the still stronger property that every subgroup of GL(n,F) has finite bounded central height. Our main result generalizes to groups of automorphisms of Noetherian modules over commutative rings.     

Class groups and Brauer groups

LetF be a global field,n a positive integer not divisible by the characteristic ofF. Then there exists a finite extensionE ofF whose class group has a cyclic direct summand of ordern. This theorem, in a slightly stronger form, is applied to determine completely, on the basis of the work of Fein and Schacher, the structure of the Brauer group Br(F()) of the rational function fieldF(t). As a consequence of this, an additional theorem of the above authors, together with a note at the end of the paper, imply that Br(F(t)) ≊ Br(F(t ₁, ···,t _n)), wheret ₁, ···,t _n are algebraically independent overF.     

A new approach to Matsumoto's theorem

We give a proof of Matsumoto's theorem on K ₂ of a field using techniques from homological algebra. By considering a complex associated to the action of GL(2, F) on P ₁(F) (F a field), we derive the Matsumoto presentation for H ₀ (F ^., H ₂(SL(2, F))) and, by considering the action of GL(n + 1, F) on P ⁿ (F), we prove the stability part of the theorem; namely, that H ₀(F ^., H ₂(SL(2, F))) is isomorphic to H ₂(SL(F)) = K ₂(F).     

On completely reducible solvable subgroups of GL(n, Δ)

Let Δ be a finite field and denote by GL(n, Δ) the group ofn×n nonsingular matrices defined over Δ. LetR⊆GL(n, Δ) be a solvable, completely reducible subgroup of maximal order. For |Δ|≧2, |Δ|≠3 we give bounds for |R| which improve previous ones. Moreover for |Δ|=3 or |Δ|>13 we determine the structure ofR, in particular we show thatR is unique, up to conjugacy. This work is part of a Ph.D. thesis done at the Hebrew University under the supervision of Professor A. Mann.     

Solvable Subgroups of Out(F n ) are Virtually Abelian

Let F _n be the free group of rank n, let Aut(F _n ) be its automorphism group and let Out(F _n ) be its outer automorphism group. We show that every solvable subgroup of Out(F _n ) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F _n ) has a finite index subgroup that lifts to Aut(F _n ).     

On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

The Division Relation: Congruence Conditions and Axiomatisability

We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y ₁,…, y _n ) involving all of the listed variables, and elements c ₁,…, c _n such that t ^A (a, c ₁,…, c _n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

Orbits of euclidean frames under discrete linear groups

We obtain certain sufficient conditions for the orbit of a (euclidean)p-frame over a vector spaceV,p<dimV, under the action of a discrete subgroup of GL(V), to be dense in the corresponding orbit of a Lie subgroup of GL(V). Using the result we classify thep-frames whose orbits under SL (n,Z) are dense in the space ofp-frames and deduce, in turn, a classification of dense orbits of certain horospherical flows. A similar result is obtained for Sp (2n,Z) forp≦n.     

On the Schur theorem for n-ary groups

We prove n-ary analogs of the well-known Schur theorem on the finiteness of a commutator subgroup of a group whose center is of finite index. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 730–741, June, 2006.     

Gauss periods as constructions of low complexity normal bases

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343–366); in particular, optimal normal bases are Gauss periods of type (n, 1) for any characteristic and of type (n, 2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type (n, t) for all n and t = 3, 4, 5 over any finite field and give a slightly weaker result for Gauss periods of type (n, 6). In addition, we give some general results on the so-called cyclotomic numbers, which are intimately related to the structure of Gauss periods. We also present the general form of a normal basis obtained by the trace of any normal basis in a finite extension field. Then, as an application of the trace construction, we give upper bounds on the complexity of the trace of a Gauss period of type (n, 3).     

A local-global principle for finiteness properties ofS-arithmetic groups over number fields

In the preceding paper [AT] compactness propertiesC _n andCP _n for locally compact groups were introduced. They generalize the finiteness propertiesF _n andFP _n for discrete groups. In this paper a local-global principle forS-arithmetic groups over number fields is proved. TheS-arithmetic group is of typeF _n , resp.FP _n , if and only if for allp inS thep-adic completionG _p of the corresponding algebraic groupG is of typeC _n resp.CP _n . As a corollary we obtain an easy proof of a theorem of Borel and Serre: AnS-arithmetic subgroup of a semisimple group has all the finiteness propertiesF _n .     

The λ- structure of the green ring Of GL(2,F p) in characteristic,IV

In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the $lambda;, and s-, structure of the Green ring of GL(2,F _p), and Sl(2,F _p). We shall now construct a subring of the Green ring which is invariant for the $lambda;, and s-, operations. It is generated by all the indecomposables with odd-dimensional composition factors. This sheds another light on the results in the previous sections. We shall also study a certain quotient of the Green ring, which is in fact the Green ring of a certain subgroup of GL(2,F _p) consisting of upper triangular matrices. The multiplication and the λ, s-, structure of this quotient Green ring is described. Moreover it is shown how this λ and s-, structure controls the deviation from being a λ, respectively s-, ring of the Green ring of any finite group with a normal Sylow subgroup of order p. The sequence of Adams operations for these groups is shown to be periodic, and the period reflects the internal p-structure of these groups.     

On division rings generated by polycyclic groups

LetD=F(G) be a division ring generated as a division ring by its central subfieldF and the polycyclic-by-finite subgroupG of its multiplicative group, letn be a positive integer and letX be a finitely generated subgroup of GL(n, D). It is implicit in recent works of A. I. Lichtman thatX is residually finite. In fact, much more is true. If charD=p≠0, then there is a normal subgroup ofX of finite index that is residually a finitep-group. If charD=0, then there exists a cofinite set π=π(X) of rational primes such that for eachp in π there is a normal subgroup ofX of finite index that is residually a finitep-group.     

Classification of the isomorphic types of martingale-H 1 spaces

LetF _n be an increasing sequence of finite fields on a probability space (Ω,F _n,P) whereF denotes the σ-algebra generated by ∪F _n. ThenF _n is isomorphic to one of the following spaces:H ¹(δ), ΣH _n ¹ ,l ^l.     

On Automorphism-Fixed Subgroups of a Free Group

Let F be a finitely generated free group, and let n denote its rank. A subgroup H of F is said to be automorphism-fixed, or auto-fixed for short, if there exists a set S of automorphisms of F such that H is precisely the set of elements fixed by every element of S; similarly, H is 1-auto-fixed if there exists a single automorphism of F whose set of fixed elements is precisely H. We show that each auto-fixed subgroup of F is a free factor of a 1-auto-fixed subgroup of F. We show also that if (and only if) n ≥ 3, then there exist free factors of 1-auto-fixed subgroups of F which are not auto-fixed subgroups of F. A 1-auto-fixed subgroup H of F has rank at most n, by the Bestvina–Handel Theorem, and if H has rank exactly n, then H is said to be a maximum-rank 1-auto-fixed subgroup of F, and similarly for auto-fixed subgroups. Hence a maximum-rank auto-fixed subgroup of F is a (maximum-rank) 1-auto-fixed subgroup of F. We further prove that if H is a maximum-rank 1-auto-fixed subgroup of F, then the group of automorphisms of F which fix every element of H is free abelain of rank at most n − 1. All of our results apply also to endomorphisms.     

On the vanishing of subspaces of alternating bilinear forms

Given a field F and integer n≥3, we introduce an invariant s_n (F) which is defined by examining the vanishing of subspaces of alternating bilinear forms on 2-dimensional subspaces of vector spaces. This invariant arises when we calculate the largest dimension of a subspace of n?×?n skew-symmetric matrices over F which contains no elements of rank 2. We show how to calculate s_n (F) for various families of field F, including finite fields. We also prove the existence of large subgroups of the commutator subgroup of certain p-groups of class 2 which contain no non-identity commutators.     

CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn (F[t]). II. FINITE FIELDS AND NUMBER FIELDS

Let K be the subgroup of SL _n (F[t]) consisting of matrices congruent to the identity modulo t. In [7] Knudson, K. 1998. Congruence Subgroups and Twisted Cohomology of SL_n(F[t]). J. Algebra, 207: 695–721.  [Google Scholar], the author conjectured that if F is a finite field, then H ₁(K) is the adjoint representation s l _n (F). A proof of this conjecture is provided in this note. The argument also works in case F is a number field. Applications to the cohomology of SL _n (F[t]) are included as is the study of the analogous question for SL _n (F[t, t ^?1]).     

Towards a Supercharacter Theory of Parabolic Subgroups

A supercharacter theory is constructed for the parabolic subgroups of the group GL(n, F_q) with blocks of orders less or equal to two. The author formulated the hypotheses on construction of a supercharacter theory for an arbitrary parabolic subgroup in GL(n, F_q).     

Donovan’s conjecture,crossed products and algebraic group actions

Donovan’s conjecture, on blocks of finite group algebras over an algebraically closed field of prime characteristicp, asserts that for any finitep-groupD, there are only finitely many Morita equivalence classes of blocks with defect groupD. The main result of this paper is a reduction theorem: It suffices to prove the conjecture for groups generated by conjugates ofD. A number of other finiteness results are proved along the way. The main tool is a result on actions of algebraic groups.