Arbitrary Finite Group Research Articles

Multidimensional polynomial patterns over finite fields: Bounds, counting estimates and Gowers norm control

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument.

On the odd order subgroups of solvable linear groups

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group.

Profinite version of the Beckmann-Black problem

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 ( K ) of some finite regular Galois extension F / K ( T ) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group.

Realizing finite groups as automizers

Abstract It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.

Entanglement asymmetry in the ordered phase of many-body systems: the Ising field theory

Global symmetries of quantum many-body systems can be spontaneously broken. Whenever this mechanism happens, the ground state is degenerate and one encounters an ordered phase. In this study, our objective is to investigate this phenomenon by examining the entanglement asymmetry of a specific region. This quantity, which has recently been introduced in the context of U(1) symmetry breaking, is extended to encompass arbitrary finite groups G. We also establish a field theoretic framework in the replica theory using twist operators. We explicitly demonstrate our construction in the ordered phase of the Ising field theory in 1+1 dimensions, where a ℤ2 symmetry is spontaneously broken, and we employ a form factor bootstrap approach to characterise a family of composite twist fields. Analytical predictions are provided for the entanglement asymmetry of an interval in the Ising model as the length of the interval becomes large. We also propose a general conjecture relating the entanglement asymmetry and the number of degenerate vacua, expected to be valid for a large class of states, and we prove it explicitly in some cases.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

The integrality of distance spectra of quasiabelian 2-Cayley graphs

A graph Γ is called an n-Cayley graph over a group G if there exists a semiregular subgroup of Aut(Γ) isomorphic to G with n orbits. In this paper, we first establish a decomposition formula for the distance spectra of n-Cayley graphs over arbitrary finite groups by using the lifted graphs, the matrix theory and the group representation theory. Secondly, we completely determine the distance splitting fields and the distance algebraic degrees of quasiabelian 2-Cayley graphs. Finally, we present some criteria for the distance integrality of quasiabelian 2-Cayley graphs. Moreover, we present some sufficient conditions for the equivalence between the integrality and the distance integrality of 2-Cayley graphs over abelian groups. We also give some sufficient conditions for the integrality of distance powers of 2-Cayley graphs over abelian groups. Our results generalize several previously known results.

New bounds for numbers of primes in element orders of finite groups

AbstractLet denote the maximal number of different primes that may occur in the order of a finite solvable group G, all elements of which have orders divisible by at most n distinct primes. We show that for all . As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.

Combinatorics of Centers of 0-Hecke Algebras in Type A

A basis of the center of the 0-Hecke algebra of an arbitrary finite Coxeter group was described by He in 2015. This basis corresponds to certain equivalence classes of the Coxeter group. We consider case of the symmetric group $S_n$. Building on work of Geck, Kim and Pfeiffer, we obtain a complete set of representatives of the equivalence classes. This set is naturally parametrized by certain compositions of $n$ called maximal. We develop an explicit combinatorial description for the equivalence classes that are parametrized by the maximal compositions whose odd parts form a hook.

Linearity of generalized cactus groups

Cactus groups are traditionally defined based on symmetric groups, and pure cactus groups are particular subgroups of cactus groups. Mostovoy [6] showed that pure cactus groups embed into right-angled Coxeter groups. We generalize this result to cactus groups associated with arbitrary finite Coxeter groups and we investigate some representations of generalized cactus groups and deduce the linearity of generalized cactus groups.

Spinning switches on a wreath product

We classify an algebraic phenomenon on several families of wreath products that can be seen as coming from a generalization of a puzzle about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we build upon a paper of Bar Yehuda, Etzion, and Moran, a paper of Ehrenborg and Skinner, and a paper of Rabinovich to provide perhaps the fullest generalization yet, modeling both the switches and the spinning table as arbitrary finite groups combined via a wreath product. We classify large families of wreath products depending on whether or not they correspond to a solvable puzzle, completely classifying the puzzle in the case when the switches behave like abelian groups, constructing winning strategies for all wreath products that are p-groups, and providing novel examples for other puzzles where the switches behave like nonabelian groups, including the puzzle consisting of two interchangeable copies of the monster group M. Lastly, we provide a number of open questions and conjectures, and provide other suggestions of how to generalize some of these ideas further.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.

On the number of edges of cyclic subgroup graphs of finite groups

In this note, we show that among finite nilpotent groups of a given order or finite groups of a given odd order, the cyclic group of that order has the minimum number of edges in its cyclic subgroup graph. We also conjecture that this holds for arbitrary finite groups.

G-Circulant Quantum Markov Semigroups

We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of [Formula: see text]-circulant GKSL generator and [Formula: see text]-circulant QMS from the circulant case, corresponding to [Formula: see text], to an arbitrary finite group [Formula: see text]. Second, we show that each [Formula: see text]-circulant GKSL generator has a block-diagonal representation [Formula: see text], where [Formula: see text] is a [Formula: see text]-circulant matrix determined by some [Formula: see text]. Denoting by [Formula: see text] the subgroup of [Formula: see text] generated by the support of [Formula: see text], we prove that [Formula: see text] has its own block-diagonal matrix representation [Formula: see text] where [Formula: see text] is an irreducible [Formula: see text]-circulant matrix and [Formula: see text] is the index of [Formula: see text] in [Formula: see text]. Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of [Formula: see text]-circulant QMSs.

Equivariant cohomology and the super reciprocal plane of a hyperplane arrangement

In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $\mathbb{Z}$-graded coefficients of localizations of ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give an interpretation of these rings in terms of superschemes, which can be used to further illuminate their structure.

Some simple biset functors

Abstract Let 𝑝 be a prime number, let 𝐻 be a finite 𝑝-group, and let 𝔽 be a field of characteristic 0, considered as a trivial F ⁢ Out ⁢ ( H ) \mathbb{F}\mathrm{Out}(H) -module. The main result of this paper gives the dimension of the evaluation S H , F ⁢ ( G ) S_{H,\mathbb{F}}(G) of the simple biset functor S H , F S_{H,\mathbb{F}} at an arbitrary finite group 𝐺. A closely related result is proved in the last section: for each prime number 𝑝, a Green biset functor E p E_{p} is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation E p ⁢ ( G ) E_{p}(G) is a free abelian group of rank equal to the number of conjugacy classes of 𝑝-elementary subgroups of 𝐺.

Finite groups, minimal bases and the intersection number

Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( G ) ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of G $G$ . In this paper, we investigate the intersection number of G $G$ , denoted α ( G ) $\alpha (G)$ , which is the minimal number of maximal subgroups whose intersection coincides with Frat ( G ) ${\rm Frat}(G)$ . In earlier work, we studied α ( G ) $\alpha (G)$ in the special case where G $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that α ( G ) ⩽ 4 $\alpha (G) \leqslant 4$ for every almost simple group G $G$ , which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups G $G$ we present best possible bounds on a related invariant β ( G ) $\beta (G)$ , which we call the base number of G $G$ . In this setting, β ( G ) $\beta (G)$ is the minimal base size of G $G$ as we range over all faithful primitive actions of the group and we prove that the bound β ( G ) ⩽ 4 $\beta (G) \leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group S a b $S_{ab}$ on the set of partitions of [ 1 , a b ] $[1,ab]$ into a $a$ parts of size b $b$ , determining the exact base size for a ⩾ b $a \geqslant b$ . This extends earlier work of Benbenishty, Cohen and Niemeyer.

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra’s character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra’s work the only information we have about these constants is that they are uniquely determined by an inductive property. Later, Goresky–Kottwitz–MacPherson (1997) and Herb (2000) gave different formulas for these constants. In this article, we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel (2011). We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Goresky– Kottwitz–MacPherson’s 1997 paper. In an appendix, we extend Herb’s notion of 2-structures to pseudo-root systems.

Lattice model constructions for gapless domain walls between topological phases

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.