Finite Group Research Articles

Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne–Michel's unique Jordan decomposition in the connected center case.

On the sum of character codegrees of finite groups

On the sum of character codegrees of finite groups

Blocks whose defect groups are Suzuki 2-groups

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.

Algebraic Enumeration of Polypolyhedra

Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order to enumerate the 3-dimensional polypolyhedra.

A new characterization of E 8(p) via its vanishing elements

Let G be a finite group, and g ∈ G . Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo ( N ) = Vo ( G ) is isomorphic to G. In this paper, we investigate the V-recognizability of E 8 ( p ) , where p is a prime number. As an application, among the 610 primes p with p < 10000 and p ≡ 0 , 1 , 4 ( mod 5 ) , we obtain that the method is always valid for confirming the V-recognizability of E 8 ( p ) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589.

Construction of Finite 2-Transitive Groups Based on the Concept of Near Field

Transitive permutation groups were studied basically based on the idea of a near field. The article mainly considered Dickson Near field to construct groups which are 2-transitive. To achieve the result, some restrictions were observed in relation to the field. The research went through the construction with the help of the classification scheme for the finite primitive group proposing that the subgroup of the group is the stabilizer of the group. The idea of this construction has relevance in coding theory.

A solvability criterion for finite groups related to vanishing conjugacy classes

A solvability criterion for finite groups related to vanishing conjugacy classes

Unramified Brauer group of quotient spaces by finite groups

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups.

On Hall normally embedded subgroups and the p-nilpotency of finite groups

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H G of H in G. In this paper, we shall obtain some characterizations about p-nilpotency of G under the assumption that certain some subgroups of G of prime power order are Hall normally embedded in G.

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

On eventually injective endomorphisms

We describe a property of linear groups that strengthens the Hopf property of finitely generated residually finite groups.

FINITE GROUPS WHOSE CHARACTER CODEGREES ARE CONSECUTIVE INTEGERS

Abstract Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers.

Polynomials Counting Nowhere-Zero Chains Associated with Homomorphisms

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ZE orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let Aψ[N] be the set of vectors g from (A−0)E, such that ∑e∈Eg(e)·f(e)=ψ(f) for each f∈N (where · is a scalar multiplication). We show that |Aψ[N]| can be evaluated by a polynomial function of |A|. In particular, if ψ(f)=0 for each f∈N, then the corresponding assigning polynomial is the classical characteristic polynomial of M.

On the strong domination number of proper enhanced power graphs of finite groups

On the strong domination number of proper enhanced power graphs of finite groups

Property FA is not a profinite property

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups \Delta and \Gamma with \Delta having Property FA, \Gamma having a non-trivial action on a tree, and \Delta and \Gamma having isomorphic profinite completions.

The formation residual of factorized finite groups

The formation residual of factorized finite groups

Clifford's theorem for bricks

Let G be a finite group, N a normal subgroup of G, and k a field of characteristic p>0. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup N in G is a p-power.

Towards a classification of simple partial comodules of Hopf algebras

Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra H using central idempotents in right coideal subalgebras and show that any 1-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial H-comodules arise this way. For H=kG for some finite group G, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If H=kG⁎, then our construction recovers the work of M. Dokuchaev and N. Zhukavets [12]. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra A.

Pro-[formula omitted] RAAGs

Let C be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-C group GΓ (pro-C RAAG for short) is the pro-C completion of the right-angled Artin group G(Γ) associated with the finite simplicial graph Γ.In the first part, we describe structural properties of pro-C RAAGs. Among others, we describe the centraliser of an element and show that pro-C RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-p subgroups of pro-C RAAGs are either free pro-p or free abelian pro-p.In the second part, we characterise splittings of pro-C RAAGs in terms of the defining graph. More precisely, we prove that a pro-C RAAG GΓ splits as a non-trivial direct product if and only if Γ is a join and it splits over an abelian pro-C group if and only if a connected component of Γ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-C RAAG, in the sense of Guirardel and Levitt [9].

Local Hilbert–Schmidt stability

We introduce a notion of local Hilbert–Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert–Schmidt stable but not Hilbert–Schmidt stable. For amenable groups, we provide a criterion for local Hilbert–Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert–Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert–Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert–Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.