On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ?, we investigate the transitivity of localization relation of the frame ? = {f _i } _i∈I and sequence ? = {e _j } _j∈G in ? with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ? _p -column decay, ? _p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ? = {f _i } _i∈I , and sequence ? = {e _j } _j∈G , with respect to the associated map a.     

Large Sets of Lattices without Order Embeddings

Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ?₀, μ can be constructed in countably many steps by passing from a cardinal λ to 2^λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {L_i: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: L_i → L_j is an order embedding, then i = j and φ is the identity map of L_i. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Pseudo-Almost Integral Elements

Let D be an integral domain with quotient field K. We define an element α ∈ K to be pseudo-almost integral over D if there is an infinite increasing sequence {s _i } of natural numbers and a nonzero c ∈ D with cα ^{s _i}  ∈ D. We investigate when a pseudo-almost integral element is almost integral or integral. We also determine the sequences {s _i } with the property that for any domain D and α ∈ K, whenever cα ^{s _i}  ∈ D for some nonzero c ∈ D, than α is actually almost integral over D.     

Indestructibility,HOD, and the Ground Axiom

Let φ₁ stand for the statement V = HOD and φ₂ stand for the Ground Axiom. Suppose T_i for i = 1, …, 4 are the theories “ZFC + φ₁ + φ₂,” “ZFC + ¬φ₁ + φ₂,” “ZFC + φ₁ + ¬φ₂,” and “ZFC + ¬φ₁ + ¬φ₂” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, A_i = _df{δ < κ∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and V_δ?T_i} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which A_i = ?? for i = 1, …, 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then V_κ?T₁, so by reflection, B₁ = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T₁} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B₁ = ??. On the other hand, assuming κ is supercompact and no cardinal δ > κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, V_δ?T₁. It is thus not possible to prove in ZFC that B_i = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T_i} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Jordan τ-derivations of Prime Rings

Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q _ms (R), and let τ be an anti-automorphism of R. An additive map δ: R → Q _ms (R) is called a Jordan τ-derivation if δ(x ²) = δ(x)x ^τ + xδ(x) for all x ∈ R. A Jordan τ-derivation of R is called X-inner if it is of the form x → ax ^τ ? xa for x ∈ R, where a ∈ Q _ms (R). It is proved that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim  _C RC = 4, where C is the extended centroid of R.     

A New Construction for Cohen–Macaulay Graphs

Let G be a finite simple graph on a vertex set V(G) = {x ₁₁,…, x _n1}. Also let m ₁,…, m _n  ≥ 2 be integers and G ₁,…, G _n be connected simple graphs on the vertex sets V(G _i ) = {x _i1,…, x _{im i} }. In this article, we provide necessary and sufficient conditions on G ₁,…, G _n for which the graph obtained by attaching the G _i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs.     

Finite Groups with Few Non-Abelian Subgroups

Let G be a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined.     

The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P ₁,…, P _m , where P _i is a Sylow p _i -subgroup of G, and g ∈ G, denote by m _𝒫(g) the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . Previously, it was shown that the properly normalized average of m _𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m _𝒫(gx) = m _𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.     

Nonsingular Equations over Groups II

Let G be a group, t an element distinct from G, and r(t) = g ₁ t ^{l ₁} …g _k t ^{l _k}  ∈ G* ?t?, where each g _i is an element of G order greater than 2, and the l _i are nonzero integers such that l ₁ + l ₂+…+l _k  ≠ 0 and |l _i | ≠ |l _j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ?G*t | r(t)? is injective. This together with previous results show that the natural map from G is injective for k ≥ 1.     

Classification of Some Graded not Necessarily Associative Division Algebras I

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group G, and have a basis {v _g |g ∈ G} as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in {1, ? 1}. We classify here those graded by an abelian group G of order |G| ≤8 with G non–isomorphic to ?/8?. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ?₀, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).     

On the Marginal Automorphisms of a Group

Let W be a nonempty subset of a free group. We call an automorphism α of a group G a marginal automorphism if x ^?1α(x) ∈ W*(G) for each x ∈ G, where W*(G) is the marginal subgroup of G. In this article, we give some results on marginal automorphisms of a group.     

On Finite p-Groups Whose IA-Automorphisms Are All Class Preserving

Let G be a finite non-abelian p-group, where p is a prime. An automorphism α of G is called a class preserving automorphism if α(x) ∈ x ^G the conjugacy class of x in G, for all x ∈ G. An automorphism α of G is called an IA-automorphism if x ^?1α(x) ∈ G′ for each x ∈ G. In this paper, we give necessary and sufficient conditions on finite p-group G of nilpotency class 2 such that every IA-automorphism is class preserving.     

Module Correspondences for Twisted Group Algebras

Abstract

Let G and A be finite groups such that (|G|, |A|) = 1. Let K be an algebraically closed field with Char K = 0. Denote by K ^α G the twisted group algebra of G over K with factor set α. In this paper we prove that if A acts homogeneously on K ^α G, then there exists an action of A on G, and there is a one-to-one correspondence between the set of A-invariant irreducible K ^α G-modules and the set of irreducible K ^α C _G (A)-modules.     

The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Abstract

Let D be a Prüfer domain, and denote by ± b?(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ? B. Denote by G _i the value group of the valuation associated with the valuation ring D _{M i} , where {M _i } _i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± b?(D) into ± b∏ _i∈I G _i is an isomorphism onto the cardinal sum ± b? _i∈I G _i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b? _i∈I G _i , the notation being as above.     

Non-Nilpotent Graph of a Group

We associate a graph 𝒩 _G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of 𝒩 _G and its induced subgraph on G \ nil(G), where nil(G) = {x ∈ G | ? x, y ? is nilpotent for all y ∈ G}. For any finite group G, we prove that 𝒩 _G has either |Z*(G)| or |Z*(G)| +1 connected components, where Z*(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of 𝒩 _G has at most two elements.     

Infeasible Mehrotra-Type Predictor-Corrector Interior-Point Algorithm for the Cartesian P *(κ)-LCP Over Symmetric Cones

We propose an infeasible Mehrotra-type predictor-corrector algorithm with a new center parameter updating scheme for Cartesian P _*(κ)-linear complementarity problem over symmetric cones. Based on the Nesterov-Todd direction, we show that the iteration-complexity bound of the proposed algorithm is 𝒪((1 + κ)³ r ²log ε^?1), where r is the rank of the associated Euclidean Jordan algebras and κ is the handicap of the problem and ε > 0 is the required precision. Some numerical results are reported as well.