We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set, and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).

The problem of constructing or characterizing strongly regular Cayley graphs (or equivalently, regular partial difference sets) has garnered significant attention over the past half-century. A classic result in this area is the complete classification of strongly regular Cayley graphs over cyclic groups, which was established by Bridges and Mena (1979), independently by Ma (1984), and partially by Marušič (1989). Miklavič and Potočnik (2003) extended this work by providing a complete characterization of distance-regular Cayley graphs over cyclic groups through the method of Schur rings. Building on this, Miklavič and Potočnik (2007) formally posed the problem of characterizing distance-regular Cayley graphs for arbitrary classes of groups. Within this framework, abelian groups are of particular significance, as many distance-regular graphs with classical parameters are Cayley graphs over abelian groups. In this paper, we employ Fourier analysis on abelian groups to establish connections between distance-regular Cayley graphs over abelian groups and combinatorial objects in finite geometry. By combining these insights with classical results from finite geometry, we classify all distance-regular Cayley graphs over the group $\mathbb{Z}_n \oplus \mathbb{Z}_p$, where $n$ is a positive integer and $p$ is an odd prime.

Abstract Tambara functors are an equivariant generalization of commutative rings. In previous work, Nakaoka introduced the spectrum of prime ideals of a Tambara functor and computed the spectrum of the Burnside Tambara functor, the equivariant analogue of the Zariski spectrum of the integers, over cyclic $p$-groups. Subsequently, Calle and Ginnett computed the spectrum of the Burnside Tambara functor over arbitrary finite cyclic groups using a generalization of Dress’ ghost coordinates for Burnside rings. In this paper, we compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates, which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_{8}$, the alternating group $A_{4}$, and the general linear group $GL_{3}(\mathbb{F}_{2})$.

Efficient tuning of structural distortion is an attractive approach for regulating self-trapped excitons emission properties of two-dimensional halide perovskites. Nevertheless, it remains elusive as to how the structural distortion is related with such emission. Here, we elucidate the relationship between structural distortion and the emission behavior of self-trapped excitons in two-dimensional lead bromide perovskites (R-NH3)2PbBr4 (where R is the cyclic carbon group). We reveal that rather than the octahedral tilting distortion, the lone pair activity-induced Jahn-Teller distortion plays a significant role in self-trapped excitons emission. Consequently, with growing ring size of cyclic organic cation, the increased Jahn-Teller distortion results in a larger relative self-trapped excitons emission due to the enhanced short-range Holstein electron-phonon coupling strength. Our work clarifies the controversy of the structural distortion correlation on self-trapped excitons emission and provides valuable insights for structural engineering in white-light emitting applications of two-dimensional perovskites.

Quaternized perfluorinated polymers are promising anion exchange membranes (AEMs) for hydrogen energy devices due to their excellent dimensional stability and high ionic conductivity, which arise from limited ion exchange capacities (IECs) and well-developed hydrophilic/hydrophobic microphase-separated morphologies. However, their poor alkaline stability remains a critical challenge. In this study, a series of Aquivion-based perfluorinated AEMs was synthesized with varied tethering structures to systematically examine the effects of sulfonyl-containing linkages, alkyl spacers, and nitrogen-based cyclic cationic end groups on alkaline stability. We specifically introduced sulfonate ester linkages and alkaline-stable cationic groups, such as N-methylpiperidinium and 1,2-dimethylimidazolium, into perfluorinated AEMs, as these functionalities have not been previously used in such systems. Compared with the sulfonyl- and sulfonate ester-linked AEMs, the sulfonamide-linked AEMs incorporating hexyl spacers exhibited markedly enhanced alkaline stability by preventing hydrolytic cleavage of the linkage between the perfluoroalkyl ether side chains and cationic end groups. Among them, the AEM bearing N-methylpiperidinium end groups (Aquivion-SO2NH-6CPip) showed greater hydroxide conductivity retention (83.8%) after immersion in 1 M KOH at 60 °C for 192 h, while the AEM containing 1,2-dimethylimidazolium end groups (Aquivion-SO2NH-6CIm) achieved a higher hydroxide conductivity of 2.56 × 10−2 S cm−1 at 80 °C. Both Aquivion-SO2NH-6CIm and Aquivion-SO2NH-6CPip demonstrated good potential in water electrolysis and fuel cell applications. Notably, the Aquivion-SO2NH-6CIm AEM exhibited good water electrolysis performance, achieving a high current density of 518 mA cm−2 at 2.2 V, which is comparable to that of the commercial Sustainion X37-RT membrane.

We study unique games and estimate some of their values. We prove that if a unique game has a quantum-assisted value close to 1, then it must have a perfect deterministic strategy. We introduce a family of unique games based on groups that generalize XOR games, and show that when the group is the cyclic group of order 3, then these games correspond to a 3-labelling problem for directed graphs.

Given a finite group G, let P(G) denote the power graph of the group G. Let Q(G) denote the signless Laplacian matrix of a graph G. Moreover, let λ1 and λn denote the largest and smallest eigenvalues of Q(G). The signless Laplacian spread of Q(G) is defined as λ1−λn. In this paper, we have described the signless Laplacian spread of the power graph of the finite cyclic group Zn. We provide the exact value of the signless Laplacian spread of the power graph of Zn when n is a power of a prime number, or when n is a product of two distinct prime numbers. For other forms of n, we provide lower and upper bounds on the same.

Cyclic groups form a foundational concept in abstract algebra, serving as essential building blocks for understanding broader group structures and algebraic systems. This paper presents a new perspective on the structure of cyclic groups by exploring their intrinsic properties through an algebraic and geometric lens. The study reinterprets the generation process, subgroup hierarchy, and element order distribution within cyclic groups, revealing novel connections between arithmetic progressions and group homomorphisms. Furthermore, it examines the implications of these structural insights for applications in number theory, coding theory, and cryptography, particularly in modular arithmetic and discrete logarithmic problems. The proposed framework not only enhances the conceptual understanding of cyclic group dynamics but also provides an alternative approach to classifying finite and infinite cyclic groups. By integrating classical theorems with new analytical tools, this work offers a unifying perspective that bridges traditional group theory with emerging computational and theoretical advancements, paving the way for future research on group symmetry and algebraic structure optimization.

This survey provides an overview of recent developments in the structure and classification of minimal varieties of associative PI-algebras equipped with graded involutions, particularly over an algebraically closed field of characteristic zero and under the action of a cyclic group of prime order.

In group theory, the cyclic group is a fundamental and seriously studied and understood classes, and is a corner stone in the study of algebraic structures. This paper studies several generalized characteristics of cyclic group with the aim of extending fundamental results and their implications within a broader context and applications to chemistry. We looked at classical properties of cyclic groups, their generation, structure and subgroup behavior. We also explore generalizations such as the decomposition of finite abelian into cyclic subgroup and their behavior under direct product constructions, and the role of cyclicity in automorphism groups and homomorphic images. Using proof-based analysis, we show that these generalized properties reveal deeper structural insights and enable a quick understanding of algebraic systems. Applications to chemistry are also discussed to highlight the practical relevance of cyclic group theory. The paper concludes the directions to future research on cyclic groups in some complex algebraic systems.

The identity-order sum graph of the finite cyclic group Z₂ₚ, or ΓI_d OS(Z₂ₚ), is a new graph structure that is presented in this study. This graph offers a cohesive framework for examining element relationships within the group by combining the properties of the identity graph and the order sum graph. In particular, if and only if x = y⁻¹, x = e, or y = e, and the sum of their orders satisfies o(x) + o(y) ≥ |Z₂ₚ|, then two vertices, x and y, in ΓI_d OS(Z₂ₚ), are adjacent. Bipartite, girth, planarity, and diameter are among the graph-theoretic properties of this graph that are examined in this paper. The findings demonstrate that ΓI_d OS(Z₂ₚ) is non-bipartite with girth 3 due to triangular cycles, planar for all primes p > 2, contains a single isolated vertex, and has diameter 2.

The power graph [Formula: see text] of a group [Formula: see text] is the simple and undirected graph with vertex set [Formula: see text] and with two distinct vertices being adjacent, whenever one of them is a positive power of the other in [Formula: see text]. The independence complex of a graph [Formula: see text], is the simplicial complex [Formula: see text] with the vertex set being that of [Formula: see text] and the simplices being the independent sets of [Formula: see text]. In this paper, we study the homotopy type of the independence complex of power graphs of cyclic groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct primes and [Formula: see text].

An analytical model of a system for reading information from a pseudorandom code scale using the window method is proposed. The model is based on the theory of linear feedback shift registers and finite Galois fields. Special attention is paid to establishing a connection between the recursive structure of binary sequences of maximum length and their algebraic representation in the space of finite fields, where primitive polynomials generate multiplicative cyclic groups.

We discover a simple construction of a four-dimensional family of smooth surfaces of general type with p g ( S ) = q ( S ) = 0 p_g(S)=q(S)=0 , K S 2 = 3 K^2_S=3 with cyclic fundamental group C 14 C_{14} . We use a degeneration of the surfaces in this family to find (complicated) explicit equations of six new pairs of fake projective planes. Our methods for finding new fake projective planes involve nontrivial computer calculations which we hope will be applicable in other settings.

We prove that torsion subgroups of groups defined by C(6) , C(4) – T(4) , or C(3) – T(6) small cancellation presentations are finite cyclic groups. This follows from a more general result on the existence of fixed points for locally elliptic (every element fixes a point) actions of groups on simply connected small cancellation complexes. We present an application concerning automatic continuity. We observe that simply connected C(3) – T(6) complexes may be equipped with a \operatorname{CAT}(0) metric. This allows us to get stronger results on locally elliptic actions in that case. It also implies that the Tits alternative holds for groups acting on simply connected C(3) – T(6) small cancellation complexes with a bound on the order of cell stabilizers.

Let p be an odd prime and G a finite group with non-abelian split metacyclic Sylow p-subgroup P ⋊ Q , where P is a cyclic group of order p n for n ≤ 2 and Q is a cyclic group of order p. Under some assumptions, we calculate a truncation of the minimal relative Q-projective resolution of the trivial module, and we give relative Q-projective covers of the simple modules for the principal p-block of G. Moreover, as for its applications, we can find information about the Brauer construction of these simple modules. Furthermore, we illustrate that the assumptions above are reasonable.

Multidimensional response mechanism of sulfamethazine migration and antibiotic resistome spread in natural riverbed driven by vertical river water level fluctuations.

On residual finiteness of graphs of free groups with cyclic edge groups

We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d ≤ 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from from the literature. Our results show that these states have a rich entanglement structure, even in the bosonic subspace.

Uniquely compatible transfer systems for cyclic groups of order pq