Finite Group Research Articles

Gassmann triples with special cycle types and applications

AbstractWe show that if one of various cycle types occurs in the permutation action of a finite group on the cosets of a given subgroup, then every almost conjugate subgroup is conjugate. As a number theoretic application, corresponding decomposition types of primes effect that a number field is determined by the Dedekind zeta function. As a geometric application, coverings of Riemannian manifolds with certain geodesic lifting behaviours must be isometric.

The characteristic subgroups of an abelian p-group

We describe the characteristic subgroups of a transitive abelian 2-group, thus completing Kaplansky's study in section 18 of his classic monograph Infinite abelian groups. This appears to be a new result even for finite abelian 2-groups.

FROBENIUS–SCHUR INDICATORS FOR PROJECTIVE CHARACTERS WITH APPLICATIONS

Abstract Let $\alpha $ be a complex valued $2$ -cocycle of finite order of a finite group $G.$ The nth Frobenius–Schur indicator of an irreducible $\alpha $ -character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $ -character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$ . A formula, involving the real $\alpha $ -regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $ -characters of G under the additional assumption that these characters are class functions.

Finite groups in which every certain maximal A-invariant subgroup is nilpotent

Let G be a finite group and assume that a finite automorphism group A acts on G, such that the orders of A and G are relatively prime. We give the structure of G if every maximal A-invariant subgroup of G of order divisible by p is nilpotent for a given prime divisor p of | G | .

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

Counting gradings on Lie algebras of block-triangular matrices

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for m∈Ns we determine the number E(−)(G,m) of isomorphism classes of elementary G-gradings on the Lie algebra UT(m)(−) of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of E(−)(G,m) and as a consequence prove that the E(−)(G,⋅) determines G up to isomorphism. We also study the asymptotic growth of the number N(−)(G,m) of isomorphism classes of G-gradings on UT(m)(−) and prove that N(−)(G,m))∼|G|E(−)(G,m).

On the properties of the lattice of τ-closed totally ω-composition formations

We study the properties of the lattice c τ ω∞ of all τ-closed totally ω-composition formations of finite groups. We prove the modularity of such a lattice of formations for any subgroup functor τ and any nonempty set ω of primes. In particular, we obtain a positive answer to the question of A. N. Skiba and L. A. Shemetkov (2000) about the modularity of the lattice c∞ L of all totally L-composition formations. We establish that the lattice c τ ω∞ is a complete sublattice of the lattice cω ∞ of all totally ω-composition formations of finite groups.

Torsion homology growth and cheap rebuilding of inner-amenable groups

We prove that virtually torsion-free, residually finite groups that are inner-amenable and non-amenable have the cheap 1-rebuilding property, a notion recently introduced by Abért, Bergeron, Frączyk and Gaboriau. As a consequence, the first \ell^{2} -Betti number with arbitrary field coefficients and log-torsion in degree 1 vanish for these groups. This extends results previously known for amenable groups to inner-amenable groups. We use a structure theorem of Tucker-Drob for inner-amenable groups showing the existence of a chain of q -normal subgroups.

Crossed Product Interpretation of the Double Shuffle Lie Algebra Attached to a Finite Abelian Group

Racinet studied a scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at N th roots of unity and constructed a group scheme attached to the situation; he also showed it to be the specialization for G=\mu_{N} of a group scheme \operatorname{\mathsf{DMR}}_{0}^{G} attached to a finite abelian group G . Then Enriquez and Furusho proved that \operatorname{\mathsf{DMR}}_{0}^{G} can be essentially identified with the stabilizer of a coproduct element arising in Racinet’s theory with respect to the action of a group of automorphisms of a free Lie algebra attached to G . We reformulate Racinet’s construction in terms of crossed products. Racinet’s coproduct can then be identified with a coproduct \widehat{\Delta}^{\mathcal{M}}_{G} defined on a module \widehat{\mathcal{M}}_{G} over an algebra \widehat{\mathcal{W}}_{G} , which is equipped with its own coproduct \widehat{\Delta}^{\mathcal{W}}_{G} , and the group action on \widehat{\mathcal{M}}_{G} extends to a compatible action of \widehat{\mathcal{W}}_{G} . We then show that the stabilizer of \widehat{\Delta}^{\mathcal{M}}_{G} , hence \operatorname{\mathsf{DMR}}_{0}^{G} , is contained in the stabilizer of \widehat{\Delta}^{\mathcal{W}}_{G} thus generalizing a result of Enriquez and Furusho [Selecta Math. (N.S.) 29 (2023), article no. 3]. This yields an explicit group scheme containing \operatorname{\mathsf{DMR}}_{0}^{G} , which we also express in the Racinet formalism.

New condensation methods with applications to the computation of Brauer character tables

Condensation is a technique that can often predict a Brauer character table of a finite group with a very high degree of confidence, but without a proof of correctness. In this paper we describe a strategy that can give such a proof. We introduce and apply two novel condensation methods: virtual tensor condensation and the condensation of bilinear forms. We illustrate our strategy and new techniques with examples taken from our computation of the 5-modular Brauer character table of the sporadic simple Lyons group.

Studying The Computational Approaches And Algorithms For Calculating The Generalized Commuting Probability Of Finite Group

The calculation of the generalized commuting probability, which quantifies the likelihood of subsets of elements commuting within a finite group, is a fundamental problem in computational group theory. This abstract presents an overview of methods and algorithms developed for efficiently computing the generalized commuting probability of finite groups. The presented approaches contribute to the advancement of computational group theory, enabling researchers and practitioners to explore and understand the structure and properties of finite groups in various mathematical and scientific domains.

Extreme Reidemeister spectra of finite groups

We extend the notions of “R∞-property” and “full (extended) Reidemeister spectrum” to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small order as well as (quasi)simple groups.

Rational noncrossing Coxeter–Catalan combinatorics

AbstractWe solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.

Boundary algebras of the Kitaev quantum double model

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.

Rigid topologies on groups

Our main result is to show that every infinite, countable, residually finite group G admits a Hausdorff group topology which is neither discrete nor precompact.

On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties

A continuous action of a finite group G on a closed orientable surface X is said to be gpnf (Gilman purely non-free) if every element of G has a fixed point on X. We prove that the biggest order μ(g), of a gpnf-action on a surface of even genus g≥2, is bounded below by 8g and that this bound is sharp for infinitely many even g as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound 8g+8 for arbitrary finite continuous actions. We also describe the asymptotic behavior of μ. We define M as the set of values of the formμ˜(g)=μ(g)g+1, and its subsets M+ and M− corresponding to even and odd genera g. We show that the set M+d, of accumulation points of M+, consists of a single number 8. If g is odd, then we prove that 4g≤μ(g)<8g. We conjecture that this lower bound is sharp for infinitely many odd g. Finally, we prove that this conjecture implies that 4 is the only element of M−d, leading to Md={4,8}.

Source algebras and cohomology algebras of block ideals of finite groups with defect groups isomorphic to extraspecial $p$-groups

Source algebras and cohomology algebras of block ideals of finite groups with defect groups isomorphic to extraspecial $p$-groups

Using Schur Rings to Produce GRRs for Dihedral Groups

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following: Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn. We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

Nonstabilizing graphs arising from group actions

Consider a finite group G which acts on itself such that the relation ∼ on G, which is defined by x ∼ y iff x is a fixed point of y under this action, is both reflexive and symmetric. Let fix ( G ) represent the set of fixed points of G, i.e., fix ( G ) = { x ∈ G | x g = x for all g ∈ G } . We associate with G a new graph using G ∖ fix ( G ) as its set of vertices, connecting vertices x , y ∈ G ∖ fix ( G ) iff x is a fixed point of y. This graph is referred to as the stabilizing graph of G and is denoted by Δ s ( G ) . Additionally, the complement of the stabilizing graph Δ s ( G ) , denoted by ∇ s ( G ) , is named the nonstabilizing graph of G. The aim of this study is to analyze various properties of the nonstabilizing graph ∇ s ( G ) and to explore how this graph-theoretical concept relates to the broader understanding of group actions.

Almost cyclic elements in primitive finite linear groups

Almost cyclic elements in primitive finite linear groups