Direct Product Of Finite Groups Research Articles

Remarks on Conjectures in Block Theory of Finite Groups

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results.

Modular Conjectures for Direct Product of Finite Groups

The representation theory of a finite group, G, is an important area of research currently. This paper studied the modular representation of finite groups, which are direct products. There are three approaches to studying this representation: the ring approach, the character approach, and the module approach. Moreover, we learned some of the important conjectures in this representation, which link a representation of a finite group and its local subgroups, which are normalizer non-trivial p-subgroups. These conjectures are the McKay conjecture, Alperin’s weight conjecture, and the ordinary weight conjecture. The main aim of the proposed paper was to investigate these conjectures of direct products, the direct summands of which satisfy these conjectures for the associated tensor product of the p-block. We obtained the results by assuming the conjectures are true. Then, we used the properties of the direct products.

On $$\tau $$-tilting finiteness of block algebras of direct products of finite groups

We discuss finiteness/infiniteness of $$\tau $$ -tilting modules over tensor products of two symmetric algebras. As an application, we consider block algebras of direct products of finite groups.

On the pronormality of subgroups of odd index in some direct products of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be pronormal in [Formula: see text] if [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text] for each [Formula: see text]. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product [Formula: see text] of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of [Formula: see text]. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text]. Thus, in this paper, we obtain a criterion of pronormality of a subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] is a prime and each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text].

Polyhedral products, flag complexes and monodromy representations

This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups Aut(Fn) and outer automorphisms of free groups Out(Fn), respectively, as well as faithful representations of products of finite groups into the linear groups SL(n,Z) and GL(n,Z). These faithful representations are realized as monodromy representations.

On the table of marks of a direct product of finite groups

We present a method for computing the table of marks of a direct product of finite groups. In contrast to the character table of a direct product of two finite groups, its table of marks is not simply the Kronecker product of the tables of marks of the two groups. Based on a decomposition of the inclusion order on the subgroup lattice of a direct product as a relation product of three smaller partial orders, we describe the table of marks of the direct product essentially as a matrix product of three class incidence matrices. Each of these matrices is in turn described as a sparse block diagonal matrix. As an application, we use a variant of this matrix product to construct a ghost ring and a mark homomorphism for the rational double Burnside algebra of the symmetric group S3.

On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups

Abstract.The prime vertex graph, Δ(X), and the common divisor graph, Γ(X), are two graphs that have been deûned on a set of positive integers X. Some properties of these graphs have been studied in the cases where either X is the set of character degrees of a group or X is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

ZL-amenability and characters for the restricted direct products of finite groups

Let G be a restricted direct product of finite groups {Gi}i∈I, and let Zℓ1(G) denote the centre of its group algebra. We show that Zℓ1(G) is amenable if and only if Gi is abelian for all but finitely many i, and characterize the maximal ideals of Zℓ1(G) which have bounded approximate identities. We also study when an algebra character of Zℓ1(G) belongs to c0 or ℓp and provide a variety of examples.

Indecomposable Projective Representations of Direct Products of Finite Groups Over a Field of Characteristic p

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G p × B a finite group, where G p is a p-group and B is a p′-group. Denote by K λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z 2(G, K*). In this article, we give necessary and sufficient conditions for K λ G to be of OTP representation type, in the sense that every indecomposable K λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K λ G p -module V and an irreducible K λ B-module W.

Direct products of finite groups as unions of proper subgroups

We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.

Reduction theorems for Noether’s problem

Let K K be any field, and G G be a finite group. Let G G act on the rational function field K ( x ( g ) : g ∈ G ) K(x(g):g\in G) by K K -automorphisms and h ⋅ x ( g ) = x ( h g ) h\cdot x(g)=x(hg) . Denote by K ( G ) = K ( x ( g ) : g ∈ G ) G K(G)=K(x(g):g\in G)^G the fixed field. Noether’s problem asks whether K ( G ) K(G) is rational (= purely transcendental) over K K . We will give several reduction theorems for solving Noether’s problem. For example, let G ~ = G × H \widetilde {G}=G\times H be a direct product of finite groups. Theorem. Assume that K ( H ) K(H) is rational over K K . Then K ( G ~ ) K(\widetilde {G}) is rational over K ( G ) K(G) . In particular, if K ( G ) K(G) is rational (resp. retract rational) over K K , so is K ( G ~ ) K(\widetilde {G}) over K K .

Automorphisms of direct products of finite groups II

Automorphisms of direct products of finite groups II

CAP-subgroups in a direct product of finite groups

If a subgroup U of a finite group G has the property that either UH=UK or U∩H=U∩K for every chief factor H/K of G, then U is said to have the cover-avoidance property in G and is called a CAP-subgroup of G. It is well known that a subgroup U of a direct product G1×G2 is determined by isomorphic sections S1 of G1 and S2 of G2 and by an isomorphism ϕ between those sections. We prove that whether U is a CAP-subgroup of G1×G2 depends on the isomorphism ϕ, but not necessarily on the sections S1 and S2. Equivalently, U is a CAP-subgroup of G1×G2 if and only if UM∩G1 is a CAP-subgroup of G1 and UN∩G2 is a CAP-subgroup of G2 for all M▪G2 and N▪G1. Consequently, subdirect subgroups and CAP-subgroups of direct factors have the cover-avoidance property.

Automorphisms of direct products of finite groups

In this paper we first find the automorphism group of the direct product of n copies of an indecomposable non-abelian group. We describe the automorphism group as matrices with entries which are homomorphisms between the n direct factors. We then use this description with a generalization of a result by Bidwell, Curran, and McCaughan on Aut (H × K), where H and K have no common direct factor, to provide structure and order theorems for an arbitrary direct product.

Modular projective representations of direct products of finite groups

Modular projective representations of direct products of finite groups

FC-Groups whose central factor group can be embedded in a direct product of finite groups

We consider the question of whether an FC-group G in which the derived subgroup [G, G] is a subgroup of a direct product of finite groups must have its central factor group G/Z(G) also embeddable in a direct product of finite groups.

Proficient presentations and direct products of finite groups

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.

Equivariant homotopy of posets and some applications to subgroup lattices

In this paper we consider the action of a finite group G on the geometric realization | CP| of the order complex CP of a poset P, on which a group G acts as a group of poset automorphisms. For special cases we give the G-homotopy type of | CP|. Moreover, we provide conditions which imply that the orbit space | CPvb/ G is homotopy equivalent to the geometric realization of the order complex over the orbit poset P G . The poset P G is the set of orbits [ x] ≔ { x g | g ϵ G} of G in P ordered by [x] ≤ [y]: → ℶg ϵ G: x g ≤ y . We apply all our results to the case P = Λ( G) 0 is the lattice of subgroups H ≠ 1, G of a finite group G. For finite solvable groups G we give the G-homotopy type of Λ( G) 0 and we show that | CΛ( G) 0|/ G and | C( Λ( G) 0/ G)| are homotopy equivalent. We do the same for a class of direct products of finite groups and for some examples of simple groups. Finally we show that for the Mathieu group G = M 12 the orbit space | CΛ( G) 0|/ G and | C( Λ( G) 0/ G)| are not homotopy equivalent.

SOME CONDITIONS FOR EMBEDDABILITY OF AN $ FC$-GROUP IN A DIRECT PRODUCT OF FINITE GROUPS AND A TORSIONFREE ABELIAN GROUP

By definition, a torsionfree abelian group belongs to the class if every -group with and is embeddable in a direct product of finite groups and a torsionfree abelian group.If is a torsionfree abelian group of rank 1, then a prime .The fundamental result of the article is the following statement. Theorem. A torsionfree abelian group belongs to the class if and only if it admits a series of pure subgroups with the following properties:(I) The quotient is of rank 1, and the set is finite, .(II) For every prime , there exists a number such that whenever .Bibliography: 9 titles.

Structure of FC-groups whose periodic parts can be embedded into direct products of finite groups

Structure of FC-groups whose periodic parts can be embedded into direct products of finite groups