Some computations in the modular representation ring of a finite group

A direct sum and tensor product of matrix games and of their equivalent linear programs are defined. The relationship of a “composed” game to the corresponding “composed” linear program is derived. Conditions are obtained for the solution of the direct sum or tensor product game to be determined from the solutions of the component games. Examples are given to show how many actual conflicts can be represented by direct sum or tensor product games.

In [C1, C2, C3, C4], B.-Y. Chen introduced the tensor product of two immersions of a given Riemannian manifold; he proved that the set of all immersions of the given manifold, provided with direct sum and tensor product, defines a commutative semiring. In [DDVV] we introduced I, the commutative semiring of all transversal immersions of all differentiable manifolds in Euclidean spaces, provided with the binary operations direct sum and tensor product. In this paper we further investigate which immersions define a subsemiring or a multiplicative subsemigroup ; in particular, we fix our attention on spherical immersions of differentiable manifolds, isometric and equivariant immersions of Riemannian manifolds and immersions of finite type. Denote by E the n-dimensional Euclidean space with Euclidean metric 〈 , 〉. The n-dimensional sphere with radius r is denoted by S(r). Let f : M → E be an immersion of a differentiable manifold in a Euclidean space. Then f is said to be transversal in a point p ∈M if and only if the position vector f(p) is not tangent to M at p, i.e. f(p) / ∈ f∗(TpM). If f is transversal in every point of M , then f shortly is called transversal. Consider two differentiable manifolds M and N of dimensions r resp. s and assume that f : M → E and h : N → E are two transversal immersions. Then the direct sum map f⊕h : M×N → E : (p, q) 7→ (f(p), h(q)) and the tensor product map f ⊗ h : M × N → E : (p, q) 7→ f(p) ⊗ h(q) are again two transversal immersions. We define a symmetric relation ∼ as follows : if f : M → E is an immersion and i : E ⊂ E is a linear isometric immersion, then ∗Supported by a research fellowship of the Research Council of the Katholieke Universiteit Leuven †Senior Research Assistant of the National Fund for Scientific Research (Belgium) ‡Research Fellow of the Research Council of the K.U.Leuven Received by the editors November 1993 Communicated by A. Warrinier AMS Mathematics Subject Classification : 53C40, 53B25, 58G25

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”

<!-- *** Custom HTML *** --> This chapter develops the basics of group theory, with particular attention to the role of group actions of various kinds. The emphasis is on groups in Sections 1–3 and on group actions starting in Section 6. In between is a two-section digression that introduces rings, fields, vector spaces over general fields, and polynomial rings over commutative rings with identity. Section 1 introduces groups and a number of examples, and it establishes some easy results. Most of the examples arise either from number-theoretic settings or from geometric situations in which some auxiliary space plays a role. The direct product of two groups is discussed briefly so that it can be used in a table of some groups of low order. Section 2 defines coset spaces, normal subgroups, homomorphisms, quotient groups, and quotient mappings. Lagrange's Theorem is a simple but key result. Another simple but key result is the construction of a homomorphism with domain a quotient group $G/H$ when a given homomorphism is trivial on $H$. The section concludes with two standard isomorphism theorems. Section 3 introduces general direct products of groups and direct sums of abelian groups, together with their concrete "external" versions and their universal mapping properties. Sections 4–5 are a digression to define rings, fields, and ring homomorphisms, and to extend the theories concerning polynomials and vector spaces as presented in Chapters I–II. The immediate purpose of the digression is to make prime fields and the notion of characteristic available for the remainder of the chapter. The definitions of polynomials are extended to allow coefficients from any commutative ring with identity and to allow more than one indeterminate, and universal mapping properties for polynomial rings are proved. Sections 6–7 introduce group actions. Section 6 gives some geometric examples beyond those in Section 1, it establishes a counting formula concerning orbits and isotropy subgroups, and it develops some structure theory of groups by examining specific group actions on the group and its coset spaces. Section 7 uses a group action by automorphisms to define the semidirect product of two groups. This construction, in combination with results from Sections 5–6, allows one to form several new finite groups of interest. Section 8 defines simple groups, proves that alternating groups on five or more letters are simple, and then establishes the Jordan–Hölder Theorem concerning the consecutive quotients that arise from composition series. Section 9 deals with finitely generated abelian groups. It is proved that "rank" is well defined for any finitely generated free abelian group, that a subgroup of a free abelian group of finite rank is always free abelian, and that any finitely generated abelian group is the direct sum of cyclic groups. Section 10 returns to structure theory for finite groups. It begins with the Sylow Theorems, which produce subgroups of prime-power order, and it gives two sample applications. One of these classifies the groups of order $pq$, where $p$ and $q$ are distinct primes, and the other provides the information necessary to classify the groups of order 12. Section 11 introduces the language of "categories" and "functors." The notion of category is a precise version of what is sometimes called a "context" at points in the book before this section, and some of the "constructions" in the book are examples of "functors." The section treats in this language the notions of "product" and "coproduct," which are abstractions of "direct product" and "direct sum."

This chapter develops the basics of group theory, with particular attention to the role of group actions of various kinds. The emphasis is on groups in Sections 1–3 and on group actions starting in Section 6. In between is a two-section digression that introduces rings, fields, vector spaces over general fields, and polynomial rings over commutative rings with identitySection 1 introduces groups and a number of examples, and it establishes some easy results. Most of the examples arise either from number-theoretic settings or from geometric situations in which some auxiliary space plays a role. The direct product of two groups is discussed briefly so that it can be used in a table of some groups of low order.Section 2 defines coset spaces, normal subgroups, homomorphisms, quotient groups, and quotient mappings. Lagrange’s Theorem is a simple but key result. Another simple but key result is the construction of a homomorphism with domain a quotient group G/H when a given homomorphism is trivial on H. The section concludes with two standard isomorphism theorems.Section 3 introduces general direct products of groups and direct sums of abelian groups, together with their concrete “external” versions and their universal mapping properties.Sections 4–5 are a digression to define rings, fields, and ring homomorphisms, and to extend the theories concerning polynomials and vector spaces as presented in Chapters I–II. The immediate purpose of the digression is to make prime fields and the notion of characteristic available for the remainder of the chapter. The definitions of polynomials are extended to allow coefficients from any commutative ring with identity and to allow more than one indeterminate, and universal mapping properties for polynomial rings are proved.Sections 6–7 introduce group actions. Section 6 gives some geometric examples beyond those in Section 1, it establishes a counting formula concerning orbits and isotropy subgroups, and it develops some structure theory of groups by examining specific group actions on the group and its coset spaces. Section 7 uses a group action by automorphisms to define the semidirect product of two groups. This construction, in combination with results from Sections 5–6, allows one to form several new finite groups of interest.Section 8 defines simple groups, proves that alternating groups on five or more letters are simple, and then establishes the Jordan-Hölder Theorem concerning the consecutive quotients that arise from composition series.Section 9 deals with finitely generated abelian groups. It is proved that “rank” is well defined for any finitely generated free abelian group, that a subgroup of a free abelian group of finite rank is always free abelian, and that any finitely generated abelian group is the direct sum of cyclic groups.Section 10 returns to structure theory for finite groups. It begins with the Sylow Theorems, which produce subgroups of prime-power order, and it gives two sample applications. One of these classifies the groups of order pq, where p and q are distinct primes, and the other provides the information necessary to classify the groups of order 12.Section 11 introduces the language of “categories” and “functors.” The notion of category is a precise version of what is sometimes called a “context” at points in the book before this section, and some of the “constructions” in the book are examples of “functors.” The section treats in this language the notions of “product” and “coproduct,” which are abstractions of “direct product” and “direct sum.”KeywordsAbelian GroupNormal SubgroupConjugacy ClassCommutative RingLinear CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper aims at treating a study on Sylow theorem of different algebraic structures as groups, order of a group, subgroups, along with the associated notions of automorphisms group of the dihedral groups, split extensions of groups and vector spaces arises from the varying properties of real and complex numbers. We must have used the Sylow theorems of this work when it's generalized. Here we discuss possible subgroups of a group in different types of order which will give us a practical knowledge to see the applications of the Sylow theorems. In algebraic structures, we deal with operations of addition and multiplication and in order structures, those of greater than, less than and so on. It is through the study of Sylow theorems that we realize the importance of some definitions as like as the exact sequences and split extensions of groups, Sylow p-subgroup and semi-direct product. Thus it has been found necessary and convenient to study these structures in detail. In situations, where it was found that a given situation satisfies the basic axioms of structure and having already known the properties of that structure. Finally, we find out possible subgroups of a group in different types of order for abelian and non-abelian cases.

A classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

Matrix algebras with direct product

(2010). Groups of Cube-Free Odd Order. The American Mathematical Monthly: Vol. 117, No. 4, pp. 363-365.

In Chapter 5 the following topics are considered: the Baer-Kaplansky theorem (Section 24); continuous and discrete isomorphisms of endomorphism rings (Section 25); endomorphism rings of groups with large divisible subgroups (Section 26); isomorphisms of endomorphism rings of mixed groups of torsion-free rank 1 (Section 27); the Corner’s theorem on split realization (Section 28); realizations for endomorphism rings of torsion-free groups (Section 29); the realization problem for endomorphism rings of mixed groups (Section 30).

Let D=F1 x F2 x... x Fn be a direct product of n free groups F1, F2, * , F* * , ox an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed, T an infinite cyclic group and F another free group. Let D x a T be the semidirect product of D and T with respect to a and (D x a T) x aXIdT F the semidirect product of D xa Tand F with respect to the automorphism x id T of D Xa T induced by a. We prove that the Whitehead group of (D xa, T) X 2xidT F and the projective class group of the integral group ring Z((D x a T) X aXidT F) are trivial. These results extend that of [3]. Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by kOZ(G). We recall the definition of semidirect product of groups and the definition of twisted group ring. For undefined terminologies used in the paper, we refer to [3] and [4]. Let oc be an automorphism of G and F a free group generated by {tA}. If w is a word in tA defining an element in F, we denote by Iwl the total exponent sum of the tA appearing in w. The semidirect product G xa F of G and F with respect to a is defined as follows: G x . F=GxF as sets and multiplication in G x . Fis given by (g, w)(g', w') = (go-lwl(g'), ww'), for any (g, w), (g', w') in G x F. In particular, if F is an infinite cyclic group T= (t) generated by t, we have the semidirect product G x a T of G and T with respect to oc. Let R be an associative ring with identity and oc an automorphism of R. Let F be a free group (or free semigroup) generated by {tA}. The otwisted group ring R,[F] of F over R is defined as follows: additively R,[F]=R[F], the group ring of F over R, so that its elements are finite linear combinations of elements in F with coefficients in R. Multiplication in R,[F] is given by (rw)(rIw')=roc-1I1(r')ww', for any rw, r'w' in R,[F]. In particular, if F is a free group (resp. free semigroup) generated by t, we Received by the editors May 25, 1973. AMS (MOS) subject classfiJcations (1970). Primary 13D15, 16A26, 18F25; Secondary 16A06, 16A54.

This paper focuses on the computation of statistical moments of strains and stresses in a random system model where uncertainty is modeled by a stochastic finite element method based on the polynomial chaos expansion. It identifies the cases where this objective can be achieved by analytical means using the orthogonality property of the chaos polynomials and those where it requires a numerical integration technique. To this effect, the applicability and efficiency of several numerical integration schemes are considered. These include the Gauss–Hermite quadrature with the direct tensor product—also known as the Kronecker product—Smolyak's approximation of such a tensor product, Monte Carlo sampling, and the Latin Hypercube sampling method. An algorithm for reducing the dimensionality of integration under a direct tensor product is also explored for optimizing the computational cost and complexity. The convergence rate and algorithmic complexity of all of these methods are discussed and illustrated with the non‐deterministic linear stress analysis of a plate. Copyright © 2007 John Wiley &amp; Sons, Ltd.

On the representation rings of quivers of exceptional Dynkin type

Let G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

This paper is concerned with studying hereditary properties of primary decompositions of torsion R[X]‐modules M which are torsion free as R‐modules. Specifically, if an R[X]‐submodule of M is pure as an R‐submodule, then the primary decomposition of M determines a primary decomposition of the submodule. This is a generalization of the classical fact from linear algebra that a diagonalizable linear transformation on a vector space restricts to a diagonalizable linear transformation of any invariant subspace. Additionally, primary decompositions are considered under direct sums and tensor product.