Representations Of Finite Groups Research Articles

On m-partite oriented semiregular representations of finite groups generated by two elements

In this paper we extend the notion of oriented regular representations in the context of m-partite oriented digraphs with an integer m≥2. A group G is said to admit an m-partite oriented semiregular representation (m-POSR for short) if there exists an m-partite oriented digraph such that its automorphism group is isomorphic to G and acts semiregularly on vertices with every part of the digraph as an orbit. In this paper, we proved that G admits an m-POSR of valency two except for several cases when G is a finite group generated by at most two elements and m≥2 is an integer.

Applications of Finite Non-Abelian Simple Groups to Cryptography in the Quantum Era

AbstractThe theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modeling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.

Matrices for finite group representations that respect Galois automorphisms

AbstractWe are given a finite group H, an automorphism $$\tau $$ τ of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group $$\langle \sigma \rangle $$ ⟨ σ ⟩ of order r, and an absolutely irreducible representation $$\rho :H\rightarrow \textsf {GL} (n,L)$$ ρ : H → GL ( n , L ) such that the action of $$\tau $$ τ on the character of $$\rho $$ ρ is the same as the action of $$\sigma $$ σ . Then the following are equivalent. $$\bullet $$ ∙ $$\rho $$ ρ is equivalent to a representation $$\rho ':H\rightarrow \textsf {GL} (n,L)$$ ρ ′ : H → GL ( n , L ) such that the action of $$\sigma $$ σ on the entries of the matrices corresponds to the action of $$\tau $$ τ on H, and $$\bullet $$ ∙ the induced representation $$\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )$$ ind H , H ⋊ ⟨ τ ⟩ ( ρ ) has Schur index one; that is, it is similar to a representation over K. As examples, we discuss a three dimensional irreducible representation of $$A_5$$ A 5 over $$\mathbb {Q}[\sqrt{5}]$$ Q [ 5 ] and a four dimensional irreducible representation of the double cover of $$A_7$$ A 7 over $$\mathbb {Q}[\sqrt{-7}]$$ Q [ - 7 ] .

On some classes of finite σ-soluble <i>PσT</i>-groups

Let X be a class of groups. Suppose that with each group G ∈ X we associate some system of its subgroups τ(G). Then τ is said to be a subgroup functor on X if the following conditions are hold: (1) G ∈ τ(G) for each group G ∈ X; (2) for any epimorphism φ: A → B, where A, B ∈ X, and for any groups H ∈ τ(A) and T ∈ τ(B) we have Hφ ∈ τ(B) and Tφ-1 ∈ τ( A). In this paper, were considered some applications of such subgroup functors in the theory of finite groups in which generalized normality for subgroups is transitive.

On gamma factors for representations of finite general linear groups

On gamma factors for representations of finite general linear groups

Representations of Finite Groups

Representations of Finite Groups

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.

Projective joint spectra and characters of representations of [formula omitted

For a tuple of square complex-valued N×N matrices A1,…,An the determinant of their linear combination x1A1+⋯+xnAn, which is called a pencil, is a homogeneous polynomial of degree N in C[x1,...xn]. Zero-set of this polynomial is an algebraic set in the projective space CPn−1. This set is called the determinantal hypersurface or determinantal manifold of the tuple (A1,...,An). It was shown in [7] that determinantal hypersurfaces contain substantial information about representations of finite Coxeter groups. Namely, if G is a non-special Coxeter group of type A,B, or D, ρ1 and ρ2 are two linear representations of G, and the determinantal hypersurfaces of images of the Coxeter generators of G under ρ1 and ρ2 coincide as divisors in the projective space, the characters of ρ1 and ρ2 are equal, and, therefore, ρ1 and ρ2 are equivalent. In [30] this result was extended in the characters part to affine Coxeter groups of types B,C, and D. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of A type are not covered by this result, as their combinatorics is quite different. In this paper we explicitly construct a finite set in A˜n having the same property. We also show that every group which is a semidirect product of a finite group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensional representation of the group, the determinantal hypersurface of images of the set determines the representation character.

A Formalization of Finite Group Theory: Part III

A Formalization of Finite Group Theory: Part III

A Formalization of Finite Group Theory: Part II

A Formalization of Finite Group Theory: Part II

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Representations of Finite Groups by Posets of Small Height

Representations of Finite Groups by Posets of Small Height

Some two-weight codes invariant under the 3-fold covers of the Mathieu groups M22 and Aut(M22)

Using an approach from finite group representation theory we construct quaternary non-projective codes with parameters [Formula: see text], quaternary projective codes with parameters [Formula: see text] and [Formula: see text] and binary projective codes with parameters [Formula: see text] as examples of two-weight codes on which a finite almost quasisimple group of sporadic type acts transitively as permutation groups of automorphisms. In particular, we show that these codes are invariant under the [Formula: see text]-fold covers [Formula: see text] and [Formula: see text], respectively, of the Mathieu groups [Formula: see text] and [Formula: see text]. Employing a known construction of strongly regular graphs from projective two-weight codes we obtain from the binary projective (respectively, quaternary projective) two-weight codes with parameters those given above, the strongly regular graphs with parameters [Formula: see text] and [Formula: see text] respectively. The latter graph can be viewed as a [Formula: see text]-[Formula: see text]-symmetric design with the symmetric difference property whose residual and derived designs with respect to a block give rise to binary self-complementary codes meeting the Grey–Rankin bound with equality.

Computational aspects of Calogero–Moser spaces

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the mathbb {C}^times -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a mathbb {Q}-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.

Orthogonal irreducible representations of finite solvable groups in odd dimension

AbstractWe prove that if is a finite irreducible solvable subgroup of an orthogonal group with odd, then preserves an orthogonal decomposition of into 1‐spaces. In particular is monomial. This generalizes a theorem of Rod Gow.

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

SOME PROPERTIES ON COPRIME GRAPH OF GENERALIZED QUATERNION GROUPS

A coprime graph is a representation of finite groups on graphs by defining the vertex graph as an element in a group and two vertices adjacent to each other's if and only if the order of the two elements is coprime. In this research, we discuss the generalized Quaternion group and its properties. Then we discuss the properties of the coprime graph over the generalized Quaternion group by looking at its Eulerian, Hamiltonian, and Planarity sides. In general, the coprime graphs of the generalized quaternion group are not Eulerian, not Hamilton, and not planar graphs. The coprime graph of the generalized quaternion group is a planar graph if for a natural number .

Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability

Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability

Minimal CSS-supplemented subgroups of finite groups

Over the years, group theorists have evaluated the impact of studying supplemented subgroups on the theory of finite groups. As a result, they have described dozens of criteria for nilpotency and solubility that depend on various supplemented embedded classes of subgroups. In keeping with this approach, we have also established a new concept of CSS-supplemented subgroups. A subgroup H of a group G is described as a CSS-supplemented subgroup in G if T ⊵ G such that G = HT and H ∩ T is SS-supplemented in G. A subgroup H of G is known as an SS-supplemented subgroup in G if there is a subgroup T of G such that HT = G and H ∩ T is SS-quasinormal in T. The SS-supplemented subgroup is expanded based on this agreement by the CSS-supplemented subgroup. By describing new criteria involving formation, we have examined the minimal CSS-subgroup of G based on the structure of the SS-supplemented subgroup that affected p-solubility and p-nilpotency.