Certain Classes Of Groups Research Articles

FACTORIZATION NUMBERS OF SOME FINITE GROUPS

AbstractFor a finite group G, let F2(G) be the number of factorizations G = AB of the group G, where A and B are subgroups of G. We compute F2(G) for certain classes of groups, including cyclic groups ℤn, elementary abelian p-groups ℤpn, dihedral groups D2n, generalised quaternion groups Q4n, quasi-dihedral 2-groups QD2n(n≥4), modular p-groups Mpn, projective general linear groups PGL(2, pn) and projective special linear groups PSL(2, pn).

A relationship between twisted conjugacy classes and the geometric invariants Ω n

A group G is said to have the property R∞ if every automorphism \({\varphi \in {\rm Aut}(G)}\) has an infinite number of φ-twisted conjugacy classes. Recent work of Goncalves and Kochloukova uses the Σn (Bieri–Neumann–Strebel–Renz) invariants to show the R∞ property for a certain class of groups, including the generalized Thompson’s groups Fn,0. In this paper, we make use of the Ωn invariants, analogous to Σn, to show R∞ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the R∞ property for BS(1, n). Moreover, we give examples for which the Ωn invariants can be used to determine the R∞ property while the Σn invariants techniques cannot.

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation

The word problem for discrete groups is well known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. As a main difference to discrete groups these groups may be generated by uncountably many generators with index running over certain sets of real numbers. We study the word problem for such groups within the Blum–Shub–Smale (BSS) model of real number computation. The main result establishes the word problem to be computationally equivalent to the Halting Problem for such machines. It thus gives the first non-trivial example of a problem complete, that is, computationally universal for this model.

More countably recognizable classes of groups

It is proved that certain classes of groups that are either (locally soluble)-by-finite rank or finite rank-by-(locally soluble) are countably recognizable.

De non ordinaria recognitione PSL2

AbstractWe establish an identification result of the projective special linear group of dimension 2 among a certain class of groups the Morley rank of which is finite.

Enumeration of groups of prime-power order

Finite group theorists have been interested in counting groups of prime-power order, as a preliminary step to counting groups of any finite order and to assist in explicitly listing such groups. In 1960, G. Higman considered when the functions giving the number of groups of prime-power order pn, for fixed n and varying p, is of a particular form, called polynomial on residue classes (PORC). The suggestion that such counting functions are PORC is known as Higman's PORC conjecture. In his 1960 paper [4] he proved that a certain class of groups of prime-power order, now called exponent-p class two groups, have counting functions that are PORC, but did not furnish explicit PORC functions.

Properly 3-realizable groups

A finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π1 (K) ≅ G whose universal cover has the proper homotopy type of a 3-manifold (with boundary). We discuss the behavior of this property with respect to amalgamated products, HNN-extensions, and direct products, as well as the independence with respect to the chosen 2-polyhedron. We also exhibit certain classes of groups satisfying this property: finitely generated Abelian groups, (classical) hyperbolic groups, and one-relator groups.

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete groupGGstates that theL2L^2-Betti numbers of a finite CW-complex with fundamental groupGGare integers ifGGis torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups ofGG. Here we establish conditions under which the Atiyah conjecture for a torsion-free groupGGimplies the Atiyah conjecture for every finite extension ofGG. The most important requirement is thatH∗(G,Z/p)H^*(G,\mathbb {Z}/p)is isomorphic to the cohomology of thepp-adic completion ofGGfor every prime numberpp. An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free. We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin’s pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does. As a consequence, if such an extensionHHis torsion-free, then the group ringCH\mathbb {C}Hcontains no non-trivial zero divisors, i.e.HHfulfills the zero-divisor conjecture. In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin’s full braid group, therefore answering question B6 on http://www.grouptheory.info. Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint “Finite group extensions and the Baum-Connes conjecture”, where for example the Baum-Connes conjecture is proved for the full braid groups.

On a certain class of groups with a combinatorial condition

On a certain class of groups with a combinatorial condition

K– andL–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by ℤ∕4

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1-->K-->G-->Q-->1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.

N-Isoclinism Classes and n-Nilpotency Degree of Finite Groups

Gallagher (1970) and Gustafson (1973) introduced the commutativity degree of a finite group. In this paper, we define the n-nilpotency degree of finite groups for n ≥ 1, and prove some results as Lescot (1995) does for a certain class of groups. In particular, it is shown that the n-isoclinism of finite groups preserves their n-nilpotency degrees. Finally, some sharper and more general upper bound than previously known is constructed for the commutativity degree of non-abelian finite groups.

On totally permutable products of finite groups

The behaviour of totally permutable products of finite groups with respect to certain classes of groups is studied in the paper. The results are applied to obtain information about totally permutable products of T, PT, and PST-groups.

Pairwise -connected Products of Certain Classes of Finite Groups#

Subgroups A and B of a finite group are said to be 𝒩-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and 𝒩-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.

Boundary behavior for groups of subexponential growth

In this paper we introduce a method for partial description of the Poisson boundary for a certain class of groups acting on a segment. As an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric (infinitely supported) measure µ of finite entropy H(µ). This implies that the entropy h(µ) of the corresponding random walk is (finite and) positive. As another application we exhibit certain discontinuity for the recurrence property of random walks. Finally, as a corollary of our results we get new estimates from below for the growth function of a certain class of Grigorchuk groups. In particular, we exhibit the first example of a group generated by a finite state automaton, such that the growth function is subexponential, but grows faster than exp(n α ) for any α< 1. We show that in some of our examples the growth function satisfies exp( n ln

Polynomially bounded cohomology and discrete groups

We establish the homological foundations for studying polynomially bounded group cohomology, and show that the natural map from PH * ( G ; Q ) to H * ( G ; Q ) is an isomorphism for a certain class of groups.

Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras

Let G be a locally compact group, Lp(G) be the usual Lp‐space for 1 ⩽ p ⩽ ∞, and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote McbA(G). We show that McbA(G) can be characterised as the ‘invariant part’ of the space of (completely) bounded normal L∞(G)‐bimodule maps on B(L2(G)), the space of bounded operators on L2(G). In doing this we develop a function‐theoretic description of the normal L∞(X, μ)‐bimodule maps on B(L2(X, μ)), which we denote by V∞(X, μ), and name the measurable Schur multipliers of (X, μ). Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to obtaining the functorial properties of McbA(G), and a concrete description of a standard predual of McbA(G). 2000 Mathematics Subject Classification 46L07, 43A30, 43A15, 46A32, 22D10, 22D12 (primary), 22D25, 43A20, 43A07 (secondary).

Totally permutable products of certain classes of finite groups

Subgroups A and B of a finite group are said to be totally permutable if every subgroup of A permutes with every subgroup of B. The behaviour of finite pairwise totally permutable products are studied with respect to certain classes of groups including the class of groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of groups where all the subnormal subgroups are permutable, the so-called PT-groups.

Approximating L2‐invariants and the Atiyah conjecture

AbstractLet G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion‐free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2‐Betti numbers over ℚG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number‐theoretic properties of eigenvalues for the combinatorial Laplacian on L2‐cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class 𝒞. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 𝒢. © 2003 Wiley Periodicals, Inc.

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K -a ndL-theory and the topolo- gicalK-theory of cocompact planar groups (D cocompact N.E.C-groups) and of groupsG appearing in an extension 1 ! Z n ! G ! ! 1w here is a finite group and the conjugation - action on Z n is free outside 0 2 Z n . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups. Mathematics Subject Classifications ( 2000): 19B28, 19D50, 19G24, 19K99.