Free subgroups in maximal subgroups of GLn(D)

Let D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GL_n(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of M_n(D) such that K*M, K∕F is Galois with Gal(K∕F)?M∕K* and Gal(K∕F) is a finite simple group.     

Length and Stable Length

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive ϵ there is a positive δ depending only on n and on ϵ such that an element of π₁(M) with stable commutator length less than δ is represented by a geodesic with length less than ϵ. Moreover, for any such M, the first accumulation point for stable commutator length on conjugacy classes is at least 1/12. Conversely, “most” short geodesics in hyperbolic 3-manifolds have arbitrarily small stable commutator length. Thus stable commutator length is typically good at detecting the thick-thin decomposition of M, and 1/12 can be thought of as a kind of homological Margulis constant. Received: June 2006 Revision: May 2007 Accepted: June 2007     

Finite A-groups with complementable nonmetacyclic subgroups

We study groups G that satisfy the following conditions (i) G is a finite solvable group with nonidentity primary metacyclic second commutator subgroup (ii) all Sylow subgroups of G are elementary Abelian. We describe the structure of groups of this type with complementable nonmetacyclic subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 607–615, May, 2006.     

Groups with finitely many normalizers of non-abelian subgroups

Abstract A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation. Keywords: normalizer subgroup, metahamiltonian group Mathematics Subject Classification (2000): 20F24     

Discrete orthogonal projections on multiple knot periodic splines

This paper establishes properties of discrete orthogonal projections on periodic spline spaces of order r, with knots that are equally spaced and of arbitrary multiplicity Mr. The discrete orthogonal projection is expressed in terms of a quadrature rule formed by mapping a fixed J-point rule to each sub-interval. The results include stability with respect to discrete and continuous norms, convergence, commutator and superapproximation properties. A key role is played by a novel basis for the spline space of multiplicity M, which reduces to a familiar basis when M=1.     

Construction of classes of subgroups of small index in p-groups

A new algorithm to calculate the conjugacy classes of subgroups of index p and p ² of a p-group is presented. It uses a surprising relation between the commutator subgroup of a p-group and linear algebra. The algorithm is by magnitudes faster and uses much less memory than the usual “Neubüser-Felsch” algorithm. However, it is restricted to this special case.     

On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Next, denote by IRR⁰(FG|M) the set of all simple FG-modules V lying over M such that the p-part of dim_F V is as small as possible. In this paper, |IRR⁰(FG|M)| and the vertices of modules in IRR⁰(FG|M) are determined. The p-blocks of G to which modules in IRR⁰(FG|M) belong are also determined.Received: 5 December 2003     

On the desking completions,theta completions and theta pairs for maximal subgroups *

Given a maximal subgroup M of a finite group G,a θ completion of M in G is any subgroup C such that C is not contained in M while M_G , the core of M in G, is contained in C and C/M_G has no propor normal subgroup of G/M_G . By using this concept we can reveal the relationship between the concepts of completions and θ-pairs introduced respectively by Deskins, Mukherjee and Bhattacharya. The concept of maximal θ-completions offers a convenience for us to study the Deskins completions in inductions. We obtain in this paper several results which imply a group to be solvable, supersolvable and nilpotent.     

Mutually permutable subgroups and Fitting classes

Let G = AB be the mutually permutable product of the subgroups A and B. It is shown that if A and B are contained in a Fitting class , then the commutator subgroup G′ of G is also contained in . Received: 14 August 2006 Revised: 17 September 2006     

Separating quasiconvex subgroups of right-angled Artin groups

A graph group, or right-angled Artin group, is a group given by a presentation where the only relators are commutators of the generators. A graph group presentation corresponds in a natural way to a simplicial graph, with each generator corresponding to a vertex, and each commutator relator corresponding to an edge. Suppose that G is a graph group whose corresponding graph is a tree and H is a subgroup of G. We show that if H is quasiconvex with respect to either the word metric on G or the CAT(0) metric on the universal cover of the standard complex for G, then H is separable, that is, H is the intersection of finite index subgroups of G. We also discuss some consequences relating to certain 3-manifold groups. Received: 19 July 2000; in final form: 2 March 2001 / Published online: 29 April 2002     

Invariant additive subgroups of a certain kind

LetR be a prime ring with a nonzero nil right ideal, and letM be the union of all nil right ideals ofR. IfW is an additive subgroup ofR which is invariant under conjugation by all special automorphisms 1+x forx ∈M, then eitherW is central orW contains a noncommutative Lie ideal ofR. Assuming thatW is invariant under only those 1+x forx ∈M andx ²=0, the same conclusion holds if the extended centroid ofR is not GF(2).     

Seshadri constants in finite subgroups of abelian surfaces

Given an étale quotient q : X → Y of smooth projective varieties we relate the simple Seshadri constant of a line bundle M on Y with the multiple Seshadri constant of q ^* M in the points of the fiber. We apply this method to compute the Seshadri constant of polarized abelian surfaces in the points of a finite subgroup.      

Boundedness of certain automorphism groups of an open manifold

It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold (being the interior of a compact manifold) are bounded. It follows that these groups are uniformly perfect. In order to characterize the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that the commutator length diameter of the automorphism group D(M){\mathcal{D}(M)} of a portable manifold M is estimated by 4.     

Polynomial identities and maximal subgroups of skew linear groups

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GL _n (D). In this paper we generalize some known results about maximal subgroups of GL _n (D) to maximal subgroups of N. More precisely, we prove that if M is a maximal subgroup of N such that F[M] satisfies a polynomial identity and contains an algebraic element over F or and either n ≥ 2 or n = 1 and M is not abelian, then [D : F] < ∞. This research was partially supported by a grant from IPM (No. 85160047).     

Linear recurring sequence subgroups in finite fields

Given a finite field and a linear recurrence relation over it is possible to find, in a fairly “obvious” way, a finite extension of and a subgroup M of the multiplicative group of such that the elements of M may be written, without repetition, so as to form a cyclically closed sequence which obeys the recurrence. Here we investigate this phenomenon for second-order recurrences; the situation in which has prime order and the characteristic polynomial of the relation is irreducible over is described.