A New Characterization of A 10 by its Noncommuting Graph

Let G be a nonabelian group and associate a noncommuting graph ?(G) with G as follows: The vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture.

Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}.

In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows.

AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M.

In this short article we prove that if G is a finite group with ?(G) ? ? (A ₁₀), then G ? A ₁₀, where A ₁₀ is the alternating group of degree 10.     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

Classification of Finite Groups with a CC-Subgroup

Abstract

A proper subgroup M of a group G is called a CC-subgroup of G if the centralizer C _G (m) of every m ∈ M ^# = M ? {1} is contained in M. In this paper we classify all finite groups containing a CC-subgroup, extending work of many authors.     

Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){\rm Fit}_{{\bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.     

Characterization of finite simple group <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2)

Let G be a finite group, and let π _e (G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M), where M = D _n (2) and n is even.     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.     

G‐invariant persistent homology

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self‐homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo‐distance d_G associated with a group G ? Homeo(X), we need to adapt persistent homology and consider G‐invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G‐invariant persistent homology with respect to the natural pseudo‐distance d_G. We also show how G‐invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self‐homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Cuspidal Modules as Summands of a Gel'fand–Graev Module

Abstract

Let G = GL _n (q) be the general linear group over a finite field 𝔽 _q with q elements. We call a Gel'fand–Graev module to be the module which affords the Gel'fand–Graev character defined in Definition I.1. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gel'fand–Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.     

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.     

Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

Projective Resolutions for Modules over Infinite Groups

We define a notion of complexity for modules over group rings of infinite groups. This generalizes the notion of complexity for modules over group algebras of finite groups. We show that if M is a module over the group ring kG, where k is any ring and G is any group, and M has f-complexity (where f is some complexity function) over some set of finite index subgroups of G, then M has f-complexity over G (up to a direct summand). This generalizes the Alperin-Evens Theorem, which states that if the group G is finite then the complexity of M over G is the maximal complexity of M over an elementary abelian subgroup of G. We also show how we can use this generalization in order to construct projective resolutions for the integral special linear groups, SL(n, ℤ), where n ≥ 2.     

Some Extensions of PSL(2, p 2) are Uniquely Determined by Their Complex Group Algebras

In Tong-Viet's, 2012 work, the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved here that some simple groups of Lie type are determined by the structure of their complex group algebras. Let p be an odd prime number and S = PSL(2, p ²). In this paper, we prove that, if M is a finite group such that S < M < Aut(S), M = ?₂ × PSL(2, p ²) or M = SL(2, p ²), then M is uniquely determined by its order and some information about its character degrees. Let X ₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results, we prove that, if G is a finite group such that X ₁(G) = X ₁(M), then G ? M. This implies that M is uniquely determined by the structure of its complex group algebra.     

Torsion-Free Space Groups, 2-Cohomology, and Scott Modules

For a subgroupCof orderpof a finite groupG, we find the summandMof thep-adic permutation module ind_C^GZ_psuch thatH²(G, M)≠0, and determine whenMis the Scott module. This is applied to the study of torsion-free space groups.     

Generic Idempotent Modules for a Finite Group

Let G be a finite group and k an algebraically closed field of characteristic p. Let F _U be the Rickard idempotent k G-module corresponding to the set U of subvarieties of the cohomology variety V _G which are not irreducible components. We show that F _U is a finite sum of generic modules corresponding to the irreducible components of V _G . In this context, a generic module is an indecomposable module of infinite length over k G but finite length as a module over its endomorphism ring.     

On abelian quantum invariants of links in 3-manifolds

From a finite abelian group G, a quadratic form onG and an element in , we define a topological invariant of a pair(M,L) where is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin-structures and related structures which appear in this context. Received May 13, 1998 / Accepted November 11, 1999 / Published online February 5, 2001     

Varieties for cohomology with twisted coefficients

Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module Ext _kG ^* (M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples. Partially supported by grants from NSF and EPSRC     

On irreducible algebras of conformal endomorphisms over a linear algebraic group

We study the algebra of conformal endomorphisms Cend ^G,G _n of a finitely generated free module M _n over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cend _n acting irreducibly on M _n generates an essential left ideal of Cend ^G,G _n if enriched with operators of multiplication on elements of H. In particular, we describe such subalgebras for the case where G is finite. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

The norm theorem for quadratic forms over a field of characteristic 2

ABSTRACT

In this paper we prove that a finite group G is isomorphic to the finite simple group L _n (q) with n ≥ 3 if and only if they have the same set of order of solvable subgroups.

     

On Modules Over Group Rings

Let M be a right module over a ring R and let G be a group. The set MG of all formal finite sums of the form ∑? _g?∈?G m _g g where m _g ?∈?M becomes a right module over the group ring RG under addition and scalar multiplication similar to the addition and multiplication of a group ring. In this paper, we study basic properties of the RG-module MG, and characterize module properties of (MG) _RG in terms of properties of M _R and G. Particularly, we prove the module-theoretic versions of several well-known results on group rings, including Maschke’s Theorem and the classical characterizations of right self-injective group rings and of von Neumann regular group rings.