Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

We classify groups G such that the unit group 𝒰 ₁(? G) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups 𝒰 ₁(KG).     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Abstract

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If 𝔛 is a class of groups, a group G is said to have 𝔛-conjugate classes of subgroups if G/Core _G (N _G (H)) ∈ 𝔛 for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite).     

Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ～ φ(x)φ(y) for all x, y ∈ G. Here ～denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ^?1. Let 𝒲₀(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲₀(G). We show that all finite irreducible Coxeter groups (except possibly E ₈) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

A New Class of Generalized Thompson's Groups and Their Normal Subgroups

For a given group G and a homomorphism ?: G → G × G, we construct groups ?_?(G), 𝒯_?(G), and 𝒱_?(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ?_Δ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).     

Engel-Type Conditions Involving Two Generalized Skew Derivations in Prime Rings

Let ? be a prime ring of characteristic different from 2, 𝒬_r the right Martindale quotient ring of ?, 𝒞 the extended centroid of ?, F, G two generalized skew derivations of ?, and k ≥ 1 be a fixed integer. If [F(r), r]_kr ? r[G(r), r]_k = 0 for all r ∈ ?, then there exist a ∈ 𝒬_r and λ ∈ 𝒞 such that F(x) = xa and G(x) = (a + λ)x, for all x ∈ ?.     

On Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G ₁ → G ₀ → G ⁰ → G ¹ → … such that the complex Hom _R (G, H) is exact for each projective left R-module H, the module Im(G ₀ → G ⁰) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.     

The permutation projective dimension of the quaternionic group

Abstract

Let R be a commutative Noetherian local Gorenstein ring with residue field k. We show that G(k), the Gorenstein injective envelope of k, is an artinian R-module, and we compute G(k) in the case where R = k[[S]] is a semigroup ring and S is symmetric. We also show that a certain subring of the endomorphism ring of G(k) is a complete local (but possibly non-commutative) ring.     

Extreme convex set functions with finite carrier: General theory

Let Ω={1,…,n} and P={X:SΩ}. A mapping e : P→R⁺ is a convex set function if e()=0 and e(S) + e(T)e(S∩T) + e(S T) for all S. TεP. The set of convex set functions for fixed Ω is a convex cone and the paper is dealing with the extreme points of the base of this cone. To this end a representation theorem is proved: every e ε ¹ can be written as e(·)=max(m¹(·)−α₁…m^t(·)−α_t), where m¹,…,m^t are measures on P and α₁,…,α_t are nonnegative reals. Given additional requirements, the representation is unique and called “canonical”. Fix H {1,…,r},|H| 2. There is a certain subsystem of sets SεP such that m^τ(S)−α_τ=e(S) (τε H}, that is, the subsystem of sets S such that m^τ(S)−α_τ(τεH) is a maximal term in the representation of e by m¹,…,m^τ and α₁,…α_t.e is called nondegenerate is these subsystems determine the measures m¹,…,m^τ uniquely and it turns out that nondegeneracy and extremality are equivalent for e ε ¹. Moreover, it is seen that nondegeneracy is closely related to a generalized version of the problem “represent a given integer λ o by means of integer weights g,…,g_r 0 via σ^r=₁ag=λ such that the integer coefficients a satisfy 0ak (=1,…,r), where k are prescribed integer bounds. Find r such representations with the additional property that the coefficients form a nonsingular matrix.” A solution to the generalized version of this number theoretical problem is given and, finally, a few examples are discussed.     

Unitary Units Satisfying a Group Identity

Let K be a nonabsolute field of characteristic p ≠ 2, G a locally finite group and KG its group algebra. Let ?: KG → KG denote the K-linear extension of an involution ? defined on G. In this article, we prove that if the subgroup 𝒰_?(KG), i.e., the ?-unitary units of KG, satisfies a group identity, then KG satisfies a polynomial identity. Moreover, in case the prime radical of KG is nilpotent, we characterize the groups G for which 𝒰_?(KG) satisfies a group identity.     

Extreme pairs for finite groups

Let G be a finite group with.O₂(G) = 1. If V is a faithful GF(2)G-module, then 𝔓 (G,V) is the set of elementary abelian 2-subgroups A of G with m(A) ? m(V/C_V(A)). A pair G,V is an extreme pair if G = < 𝔓 (G,V)>. Such pairs often appears in analysis of 2-local subgroups of characteristic 2-type groups. The following theorem is the main result of this paper.     

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z ⁿ (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞_𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D _· = … → D _?1 → D ₀ → D ₁ → … with each term D _i in 𝒞_𝒲 such that C = ker(D ₀ → D ₁) and D _· is both Hom(𝒞_𝒲, ?) and Hom(?, 𝒞_𝒲) exact. In this article, we show that C = … → C ^?1 → C ⁰ → C ¹ → … is a completely 𝒲-resolved complex if and only if C ⁿ is a completely 𝒲-resolved module for all n ∈ ?. Some known results are obtained as corollaries.     

On c -Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ?.

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ?.     

SUBDIRECTLY IRREDUCIBLE LEFT SELF DISTRIBUTIVELY GENERATED ALGEBRAS

Abstract

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R _Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R _Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R _Δ(F) is solvable.     

Some Results on Hypercentral Units in Integral Group Rings

Abstract

In this note we investigate the hypercentral units in integral group rings ?G,where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z ₂(𝒰) is not contained in C _𝒰(T),then T is either an Abelian group of exponent 4 or a Q* group. This extends our earlier result on torsion group rings.