We demonstrate that the chiral Z_{p} toric code-the quintessential model of topological order-hosts additional, emergent topological phases when perturbed: descendant fractional quantum Hall-like states, which we term Hall-on-Toric. These hierarchical states feature fractionalized Z_{p} charges and increased topological ground-state degeneracy. The Hall-on-Toric phases appear in the vicinity of the transitions between deconfined Z_{p} phases with different background charge per unit cell, in a fixed nontrivial flux background. We confirm their existence through extensive infinite density matrix renormalization group (iDMRG) simulations, analyzing the topological entanglement entropy, entanglement spectra, and a generalized Hall conductance. Remarkably, the Hall-on-Toric states remain robust even in the absence of U(1) symmetry. Our findings reinforce the foundational interpretation of star and plaquette defects as magnetic and electric excitations, and reveal that this perspective extends to a much deeper level.

We provide a general structural criterion implying that a group has infinite m-almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite m-almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens previously known results in the literature.

We study monoids in Z ( N ) that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local–global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local–global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.

On Ulam widths of finitely presented infinite simple groups

The study of automorphisms of algebraic structures plays a central role in understanding their internal symmetries and structural behavior. This work investigates the automorphism structure induced by <i>finite subgroups within infinite groups</i>, with particular emphasis on how these automorphisms can be characterized, classified, and effectively utilized. The focus is on the interaction between a finite subgroup and the ambient infinite group, analyzing how subgroup-preserving automorphisms extend to global automorphisms and how constraints imposed by finiteness influence the overall automorphism group. Special attention is given to classes of infinite groups such as abelian, conjugacies, and certain residually finite groups where finite subgroup automorphisms exhibit rich and tractable behavior. Building on this theoretical framework, this work explores <i>applications to symmetric cryptography,</i> where algebraic symmetry and complexity are essential for secure cryptographic design. Finite subgroup automorphisms are shown to provide a promising foundation for constructing cryptographic primitives, including key generation mechanisms, conjugacy-based encryption schemes, and secure mixing transformations. The inherent difficulty of reversing automorphism actions in large infinite groups, combined with the controlled structure of finite subgroups, offers a balance between computational efficiency and cryptographic strength. In overall, this work bridges abstract group theory and practical cryptographic applications, demonstrating that finite subgroup automorphisms of infinite groups constitute a viable and mathematically robust framework for advancing symmetric cryptographic systems.

Abstract Let $${\mathcal {F}} \, = \, (\dots {\mathop {\rightarrow }\limits ^{\partial _{n+1}}} {\mathcal {F}}_n {\mathop {\rightarrow }\limits ^{\partial _n}} {\mathcal {F}}_{n-1}{\mathop {\rightarrow }\limits ^{\partial _{n-1}}} \dots \dots {\mathop {\rightarrow }\limits ^{\partial _1}} {\mathcal {F}}_0 \rightarrow {\mathfrak {R}} \rightarrow 0)$$ F = ( ⋯ → ∂ n + 1 F n → ∂ n F n - 1 → ∂ n - 1 ⋯ ⋯ → ∂ 1 F 0 → R → 0 ) be a free resolution over the group ring $${\mathfrak {R}}[\Phi ]$$ R [ Φ ] where $${\mathfrak {R}}$$ R is commutative and $$\Phi $$ Φ is finite. The $$n^{th}$$ n th syzygy $$\Omega _n^{{\mathfrak {R}}[\Phi ]}$$ Ω n R [ Φ ] is the stable class of $$\textrm{Im}(\partial _n)$$ Im ( ∂ n ) and has a tree structure with roots which do not extend infinitely downwards. We show that $$\Omega _3^{{\mathfrak {R}}[Q_{8p}]}$$ Ω 3 R [ Q 8 p ] has infinitely many isomorphically distinct modules at the minimal level when $$\,{\mathfrak {R}} = {\mathbb {Z}}[C_\infty ]$$ R = Z [ C ∞ ] is the integral group ring of the infinite cyclic group and $$Q_{8p}$$ Q 8 p is the quaternion group of order 8 p where $$p \ge 3$$ p ≥ 3 is prime. This poses severe difficulties in attempting to solve the D (2) problem of CTC Wall for the groups $$C_\infty \times Q_{8p}$$ C ∞ × Q 8 p

Abstract In an elementary way, we construct a family of pairwise non-embeddable torsion-free groups which contains 2 κ 2^{\kappa} groups of cardinality 𝜅 for every infinite cardinal number 𝜅 up to the first strong limit cardinal of uncountable cofinality.

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 main purpose of this paper is to strengthen our understanding of sofic mean dimension of two typical classes of sofic group actions. First, we study finite group actions. We prove that sofic mean dimension of any amenable group action does not depend on the choice of sofic approximation sequences. Previously, this result was known only if the acting group is an infinite amenable group. Moreover, we investigate the full shifts, for all sofic groups and all alphabets. We show that sofic mean dimension of any full shift depends purely on its alphabet. Our method is a refinement of the classical technique in relation to the estimates from above and below, respectively, for mean dimension of some typical actions. The key point of our results is that they apply to all compact metrizable spaces without any restriction (in particular, the alphabet concerned in a full shift and the space involved in a finite group action are not required to be finite-dimensional). Furthermore, we improve the quantitative knowledge of sofic mean dimension, restricted to finite-dimensional compact metrizable spaces, for those two typical classes of sofic group actions. As a direct consequence of the main ingredient of our proof, we obtain the exact value of sofic mean dimension of all the actions of finite groups on finite-dimensional compact metrizable spaces. Previously, only an upper bound for these actions was given. Besides, we also get the exact value of sofic mean dimension of full shifts when the alphabet is finite-dimensional.

Abstract We introduce the notion of Banach metrics on finitely generated infinite groups. This extends the notion of a Cayley graph (as a metric space). Our motivation comes from trying to detect the existence of virtual homomorphisms into $$\mathbb {Z}$$ Z . We show that detection of such homomorphisms through metric functional boundaries of Cayley graphs isn’t always possible. However, we prove that it is always possible to do so through a metric functional boundary of some Banach metric on the group.

Abstract Let M be an open (complete and non-compact) manifold with $$\textrm{Ric}\ge 0$$ Ric ≥ 0 and escape rate not 1/2. It is known that under these conditions, the fundamental group $$\pi _1(M)$$ π 1 ( M ) has a finitely generated torsion-free nilpotent subgroup $$\mathcal {N}$$ N of finite index, as long as $$\pi _1(M)$$ π 1 ( M ) is an infinite group. We show that the nilpotency step of $$\mathcal {N}$$ N must be reflected in the asymptotic geometry of the universal cover $$\widetilde{M}$$ M ~ , in terms of the Hausdorff dimension of an isometric $$\mathbb {R}$$ R -orbit: there exist an asymptotic cone ( Y , y ) of $$\widetilde{M}$$ M ~ and a closed $$\mathbb {R}$$ R -subgroup L of the isometry group of Y such that its orbit Ly has Hausdorff dimension at least the nilpotency step of $$\mathcal {N}$$ N . This resolves a question raised by Wei and the author (see Pan and Wei in Geom Funct Anal 32:676–685, 2022, Remark 1.7 and Pan in Geom Topol 28:1409–1436, 2024, Conjecture 0.2).

Abstract We construct an explicit infinite family of pairwise non‐isomorphic infinite simple groups of type (in particular, they are finitely presented) that act faithfully on the circle by orientation‐preserving homeomorphisms, but that admit neither non‐trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest‐skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.

Witte Morris showed in [Discrete Math, 38.1 (1982)] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product [Formula: see text]. We show that every connected Cayley graph of the free product with amalgamation [Formula: see text] has a Hamiltonian double ray, where [Formula: see text] is a generalized quasi-dihedral group on [Formula: see text] Additionally, this leads to the conclusion that each connected Cayley graph of the infinite dihedral group also contains a Hamiltonian double ray.

Consider a finite group and let denote a prime number. A Sylow subgroup of is a subgroup of whose order is as large as is allowed by Lagrange’s theorem, and is the set all such subgroups. The essential theorem of group theory asserts that Sylow subgroups always exist and mod . In this note, we say that an ordered pair is a Sylow pair if there exists group with , where is an integer. We prove that the ordered pair (7,15) is not a Sylow pair.

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.

Fractional Chern insulator states in an isolated flat band of zero Chern number

Abstract We show that every free continuous action of a countably infinite elementary amenable group on a finite-dimensional compact metrizable space is almost finite. As a consequence, the crossed products of minimal such actions are ${\mathcal{Z}}$ -stable and classified by their Elliott invariant.

Abstract Although normality is not a transitive relation in group theory, one often needs to deal with sections of a group in which this is the case. This led many prominent group theorists, such as Gaschütz and Robinson, to study (finite and infinite) groups in which normality is transitive (the so-called $$T\hbox {-}groups$$ T - g r o u p s ). The aim of this paper is to describe (mostly periodic soluble) linear groups in which normality of Zariski closed subgroups is transitive (we call this property the $$T_c\hbox {-}property$$ T c - p r o p e r t y ), so to obtain a more general framework from which some of the well-known and relevant results concerning finite soluble T -groups can be derived. Our main theorems show among other things that a soluble periodic linear group with $$T_c$$ T c -property is metabelian, hypercyclic and abelian-by-finite (see Theorem A), and that the $$T_c$$ T c -property is inherited by finite-index subgroups (see Theorem B). Also, we show that for (affine) algebraic groups, the T -property coincides with the $$T_c$$ T c -property (see Theorem D). Note that some of our results are carried out in a more general context by studying a subgroup $$\omega _c(G)$$ ω c ( G ) measuring the distance of a linear group G from having the $$T_c$$ T c -property. The relationship of $$\omega _c(G)$$ ω c ( G ) with the Wielandt subgroup is studied, and many examples are provided to show how different the behaviour of a linear group with property $$T_c$$ T c can be from that of a T -group.

Let [Formula: see text] be a group, and let [Formula: see text] be an automorphism of [Formula: see text]. If [Formula: see text] then [Formula: see text] is said to be a commuting automorphism. The set of all such automorphisms is denoted by [Formula: see text]. This set does not necessarily form a subgroup of the automorphism group of [Formula: see text]. If [Formula: see text] does form a subgroup, then [Formula: see text] is said to be an [Formula: see text]-group. Let [Formula: see text] be a set of prime numbers. Define [Formula: see text] as the ring consisting of all rational numbers [Formula: see text], where [Formula: see text] and [Formula: see text] are coprime integers, and [Formula: see text] is a [Formula: see text]-number. The additive group of [Formula: see text] is denoted by [Formula: see text]. Now let [Formula: see text] and [Formula: see text] be two sets of primes, and let [Formula: see text] be a nonzero integer. Consider a generalized extraspecial [Formula: see text]-group [Formula: see text], defined as follows: [Formula: see text] Let [Formula: see text], where [Formula: see text] is a generalized extraspecial [Formula: see text]-group such that [Formula: see text] with [Formula: see text]. In this paper, we show that if [Formula: see text], then [Formula: see text] is a non-[Formula: see text]-group, and if [Formula: see text], then [Formula: see text] is an [Formula: see text]-group. As a consequence, we identify the conditions determining when the following groups are [Formula: see text]-groups or not: (i) the direct product of a generalized extraspecial [Formula: see text]-group and a free abelian group with finite rank, (ii) an extension of [Formula: see text] by a direct sum of finitely many copies of [Formula: see text], where [Formula: see text] is the additive group of rational numbers, (iii) an infinite Černikov [Formula: see text]-group which is non-abelian but each proper quotient group is abelian.

Finitely generated infinite torsion groups that are residually finite simple