Non-abelian Group Research Articles

Integral Cayley graphs over a group of order 6n

In this paper, we study the integral Cayley graphs over a non-abelian group U 6 n = ⟨ a , b ∣ a 2 n = b 3 = 1 , a − 1 ba = b − 1 ⟩ of order 6 n . We give a necessary and sufficient condition for the integrality of Cayley graphs over U 6 n . We also study relationships between the integrality of Cayley graphs over U 6 n and the Boolean algebra of cyclic groups. As applications, we construct some infinite families of connected integral Cayley graphs over U 6 n .

Representations of extensions of simple groups

Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group G factors through a projective representation of G, except for some groups of Lie type in characteristic 2; the exact exceptions for G were determined by Kleidman and Liebeck (1989). We generalise this result in two ways. First we consider all low-dimensional projective representations, not just those of minimal dimension. Second we consider all characteristically simple groups, not just simple groups.

First-order sentences in random groups II: $\forall\exists$-sentences

We prove that a random group, in Gromov’s density model with d<1/16 , satisfies with overwhelming probability a universal-existential first-order sentence \sigma (in the language of groups) if and only if \sigma is true in a nonabelian free group. It is remarked that one can also obtain the optimal result, that is, replace d<1/16 by d<1/2 .

Phases of matrix product states with symmetric quantum circuits and symmetric measurements with feedforward

Two matrix product states (MPS) are in the same phase in the presence of symmetries if they can be transformed into one another via symmetric short-depth circuits. We consider how symmetry-preserving measurements with feedforward alter the phase classification of MPS in the presence of global on-site symmetries. We demonstrate that, for all finite Abelian symmetries, any two symmetric MPS belong to the same phase. We give an explicit protocol that achieves a transformation between any two phases and that uses only a depth-two symmetric circuit, a single round of symmetric measurements, and a constant number of auxiliary systems per site. In the case of non-Abelian symmetries, symmetry protection prevents one from deterministically transforming symmetry-protected topological (SPT) states to product states directly via measurements, thereby complicating the analysis. Nonetheless, we provide protocols that allow for asymptotically deterministic transformations between the trivial phase, SPT phases, and GHZ phases of some non-Abelian nilpotent groups. Published by the American Physical Society 2025

A random Hall-Paige conjecture

Abstract A complete mapping of a group $G$ G is a bijection $\phi \colon G\to G$ ϕ : G → G such that $x\mapsto x\phi (x)$ x ↦ x ϕ ( x ) is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ G has a complete mapping whenever $\prod _{x\in G} x$ ∏ x ∈ G x is the identity in the abelianization of $G$ G . This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. In this paper, we give a combinatorial proof of a far-reaching generalisation of the Hall-Paige conjecture for large groups. We show that for random-like and equal-sized subsets $A$ A , $B$ B , $C$ C of a group $G$ G , there exists a bijection $\phi \colon A\to B$ ϕ : A → B such that $x\mapsto x\phi (x)$ x ↦ x ϕ ( x ) is a bijection from $A$ A to $C$ C whenever $\prod _{a\in A} a \prod _{b\in B} b=\prod _{c\in C} c$ ∏ a ∈ A a ∏ b ∈ B b = ∏ c ∈ C c in the abelianization of $G$ G . We use this statement as a black-box to settle the following old problems in combinatorial group theory for large groups. (1) We characterise sequenceable groups, that is, groups which admit a permutation $\pi $ π of their elements such that the partial products $\pi _{1}$ π 1 , $\pi _{1}\pi _{2}$ π 1 π 2 , $\pi _{1}\pi _{2}\cdots \pi _{n}$ π 1 π 2 ⋯ π n are all distinct. This resolves a problem of Gordon from 1961 and confirms conjectures made by several authors, including Keedwell’s 1981 conjecture that all large non-abelian groups are sequenceable. We also characterise the related $R$ R -sequenceable groups, addressing a problem of Ringel from 1974. (2) We confirm in a strong form a conjecture of Snevily from 1999 by characterising large subsquares of multiplication tables of finite groups that admit transversals. Previously, this characterisation was known only for abelian groups of odd order (by a combination of papers by Alon and Dasgupta-Károlyi-Serra-Szegedy and Arsovski). (3) We characterise the abelian groups that can be partitioned into zero-sum sets of specific sizes, solving a problem of Tannenbaum from 1981. This also confirms a recent conjecture of Cichacz. (4) We characterise harmonious groups, that is, groups with an ordering in which the product of each consecutive pair of elements is distinct, solving a problem of Evans from 2015.

A small serving of mash: (Quantum) algorithms for SPDH-Sign with small parameters

Abstract We find an efficient method to solve the semidirect discrete logarithm problem (SDLP) over finite nonabelian groups of order p 3 {p}^{3} and exponent p 2 {p}^{2} for certain exponentially large parameters. This implies an attack on SPDH-Sign, Pronounced “SPUD-Sign”. a signature scheme based on the SDLP, for such parameters. In particular, SDLP instances over such groups are parameterised by an n < ( p − 1 ) p 6 n\lt \left(p-1){p}^{6} : we develop a method to solve instances when n ≤ poly ( log p ) ⋅ p n\le {\rm{poly}}\left(\log p)\hspace{0.25em}\cdot p . Letting λ \lambda be the security parameter of SPDH-Sign, which is taken p = exp λ p=\exp \lambda , we find we may solve instances of SDLP corresponding to SPDH-Sign instances with exponentially large p p . However, for n ≈ p 2 n\approx {p}^{2} and larger, our method no longer completely solves the SDLP instances. We also study the linear hidden shift problem for a group action corresponding to SDLP and take a step towards proving the quantum polynomial time equivalence of SDLP and the semidirect computational Diffie–Hellman problem.

The Terwilliger algebras of the group association schemes of two non-abelian groups

Recently, Maleki determined the dimension of the Terwilliger algebra of C n ⋊ C 2 , where C n ⋊ C 2 = 〈 a , b | a n = b 2 = 1 , ba b − 1 = a s 〉 . Subsequently, the authors determined the dimension of the Terwilliger algebra of T n , s , where T n , s = 〈 a , b | a 2 n = 1 , a n = b 2 , ba b − 1 = a s 〉 . In this paper, generalizing these two results, we consider the Terwilliger algebras of the generalized dihedral groups and non-abelian finite groups admitting a cyclic subgroup of index 2. Moreover, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of these groups.

Multi-toric geometries with larger compact symmetry

ABSTRACT We study complete, simply connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $ \mathrm{Spin}(7) $-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $ T^{3} \times \mathrm{SU}(2) $. We then specialize the analysis to holonomy G2, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $ \mathbb{R} \times \mathrm{SU}(2) $, and their extensions over singular orbits, to give local examples in the $ \mathrm{Spin}(7) $-case that have singular orbits where the stabilizer is of rank one.

Properties of commuting graphs over finite non-abelian groups

Properties of commuting graphs over finite non-abelian groups

Trim turnpikes for optimal control problems with symmetries

Abstract Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem.

The codegree isomorphism problem for finite simple groups II

ABSTRACT Let H be a nonabelian finite simple group. Huppert’s conjecture asserts that if G is a finite group with the same set of complex character degrees as H, then $G\cong H\times A$ for some abelian group A. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if G has the same set of character codegrees as H, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [15], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.

Bent Functions on Finite Nonabelian Groups and Relative Difference Sets

ABSTRACTIt is well known that the perfect nonlinearity of a function between finite groups and can be characterized by its graph in terms of relative difference set in the direct product (cf. [4]). Let be the infinite set of complex roots of unity. A ‐valued function on an arbitrary finite group is associated with a finite cyclic subgroup in the multiplicative group of nonzero complex numbers. For a bent function on in general, its graph is not a relative difference set in the direct product . In this paper, we investigate the necessary and sufficient conditions under which the graph of a bent function on is a relative difference set in . Cyclotomic fields and their integral bases play an important role in our discussions.

Hyers–Ulam stability of norm-additive functional equations via (δ,ϵ)-isometries

This research examines the Hyers–Ulam stability of norm-additive functional equations expressed as ∥ξ(gh−1)∥=∥ξ(g)−ξ(h)∥,∥ξ(gh)∥=∥ξ(g)+ξ(h)∥, through the utilization of (δ,ϵ)-isometries. In this context, ξ:G→X represents a surjective (δ-surjective) mapping, where G denotes noncommutative group (arbitrary group) and X signifies a Banach space (or a real Banach space).

Solvable groups with four supercharacter theories

The aim of this paper is to identify the solvable groups with four supercharacter theories. To achieve this, the research will first identify abelian groups with four supercharacter theories. Then it is proved that the only non-abelian group with four supercharacter theories is dihedral group of order 38.

Applications of Harmonic Analysis in Quantum Computing

The quantum Fourier transformation has been demonstrated to be a useful tool in dealing with quantum-mechanical problems. This paper discusses how quantum Fourier transformation is applied in finding the period of functions within Shor’s algorithm, allowing the algorithm to be much more efficient in factorizing large integers than classic algorithms. This paper also discusses how Grover’s algorithm uses the concept of harmonic process, where amplitude amplification is likened to harmonic oscillation for improving the efficiency of searching unsorted data base. The paper further explores the application of harmonic analysis in quantum error correction, particularly in stabilizer codes. The paper also introduces wavelet analysis as an alternative approach for detecting and correcting localized quantum errors. Finally, the paper extends the discussion into the potential future directions of harmonic analysis in quantum computing, such as extending Fourier transformation to non-abelian groups and using spherical harmonics in quantum algorithms which may optimize current quantum algorithms and solve currently unsolved quantum questions.

The Szeged Index Energy of Commuting Graph of Certain Finite Non-commutative Groups

Let G be a finite group having center Z(G). The commuting graph of G denoted by C(G) has vertex set as G \ Z(G), and two vertices x and y are adjacent in C(G) if x and y commute with each other. The commuting graph of a finite group is a powerful tool in group theory to understand the internal structure of the group. Through its graph-theoretic properties, the commuting graph helps in classifying groups and understanding how the group works internally. It serves as a visual and computational tool to complement other algebraic methods in group theory. The Szeged index is a topological index used in the study of molecular structures, particularly in chemistry and chemical graph theory. It is a numerical value that characterizes the connectivity of a molecular graph. In this paper, we have determined the Szeged index of the commuting graph of various finite non-commutative groups, namely the dihedral group Dn, and the dicyclic group Dicn. Moreover, we determine the energy of C(Dn) and C(Dicn). A graph is said to be hyperenergetic if the energy of G is greater than the complete graph. In this paper, we prove that the graphs C(Dn) and C(Dicn) are non-hyperenergetic graphs.

Finite $p$-groups in which the normalizer of each nonnormal subgroup is small

Let $G$ be a finite non-Dedekindian $p$-group which satisfies $N_G(H)=HZ(G)$ for each nonnormal subgroup $H$, and we call it an $NS$-group. In this paper, it is proved that an $NS$-group is the product of a minimal nonabelian group and the center.

P-Nilpotent Maximal Subgroups in Finite Groups

Let p be a prime number and suppose that every maximal subgroup of a finite group is either p-nilpotent or has prime index. Such a group need not be p-solvable, and we study its structure by proving that only one nonabelian simple group of order divisible by p, which belongs to the family PSLn(q), can be involved in it. For p=2, we specify more, and in fact, such a simple group must be isomorphic to PSL2(ra) for certain values of the prime r and the parameter a.

The axion is going dark

In this work we explore the effect of a non-Abelian dark sector gauge group that couples to the axion field. In particular, we analyze effects that arise if the dark sector gauge group mixes topologically with the Standard Model. This is achieved by gauging a subgroup of the global center 1-form symmetries which embeds non-trivially in both the visible as well as the dark sector gauge groups. Leaving the local dynamics unchanged, this effect modifies the quantization conditions for the topological couplings of the axion, which enter the estimate of a lower bound on the axion-photon coupling. In the presence of a dark sector this lower bound can be reduced significantly, which might open up interesting new parameter regions for axion physics. We further determine the allowed exotic matter representations in the presence of topological mixing with a dark sector, explore the generalized categorical symmetries of such axion theories, and comment on other model-independent phenomenological consequences.

On Commuting Automorphisms of a Finite Nonabelian Group with Metacyclic Central Quotient

On Commuting Automorphisms of a Finite Nonabelian Group with Metacyclic Central Quotient