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

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).

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.

The Levi class L(M) generated by the class of groups M is the class of all groups in which the normal closure of each cyclic subgroup belongs to M.Let p be a prime number, p ̸= 2, s be a natural number, s ≥ 2, and s > 2 for p = 3; Hps be a free group of rank 2 in the variety of nilpotent groups of class ≤ 2 of exponent ps with commutator subgroup of exponent p; Z is an infinite cyclic group; q{Hps , Z} is a quasivariety generated by the set of groups {Hps , Z}. We find a basis of quasi-identities of the Levi class L(q{Hps , Z}) and establish that there exists a continuous set of quasivarieties K such that L(K) = L(q{Hps , Z}).

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum A-flows, which gives edge weights using nonzero elements of an additive Abelian group A and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: A=Zk, the finite cyclic group of integer congruence modulo k, and A=Z, the infinite cyclic group of integers. The constant sum under a constant-sum A-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by Ik(G) and I(G) for A=Zk and A=Z, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases Z and Zk have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs Wn and fan graphs Fn in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning Zk and Z, respectively. Some open problems will be mentioned in the concluding remarks.

We prove that higher-dimensional Thompson's groups have linear divergence functions.By the work of Drut , u, Mozes, and Sapir, this implies none of the asymptotic cones of nV has a cut-point. IntroductionThompson's groups F, T , and V are finitely presented infinite groups defined by Richard Thompson in the 1960s.They are all known to be mysterious groups.For example, T and V are the first examples of finitely presented, infinite, and simple, and it is known that the amenability of F is a difficult open problem.Because they have several unpredictable properties, by focusing on such properties, many "generalized" Thompson's groups were also defined.Higher-dimensional Thompson's groups, denoted by 2V, 3V, . . ., are some such groups defined by Brin [3].The group V acts on the Cantor set C, and the group nV acts on the powers of the Cantor set C n .It is known that nV is also finitely presented [4; 14] and simple [3; 5].In addition, it was shown that for n, m >0 , the group nV is isomorphic to mV if and only if n = m holds [1].In [16], it was proved that nV has Serre's property FA, and hence is one-ended.In 2018, Golan and Sapir showed that the divergence functions of F, T , and V are linear [12].This function was first mentioned by Gromov [13], and later, Gersten gave the formal definition [11] as a quasi-isometric invariant of geodesic metric spaces.Roughly speaking, the order of the function indicates whether the Cayley graphs of the group are "close" to the Euclidean or hyperbolic spaces.In fact, the orders of the functions of the direct powers of the infinite cyclic group 2 , 3 , . . .are linear, and it is known that the orders of the functions of hyperbolic groups are at least exponential [2].In [12], they asked whether their proof could be extended to generalized Thompson's groups.In recent years, similar results have been obtained for some groups by extending the original arguments [17; 19; 18].For recent results on functions of groups other than generalized Thompson's groups, see [15].

As part of their construction of the Khovanov spectrum, Lawson, Lipshitz and Sarkar assigned to each cube in the Burnside category of finite sets and finite correspondences, a finite cellular spectrum. In this paper, we extend this assignment to cubes in Burnside categories of infinite sets. This is later applied to the work of Akhmechet, Krushkal and Willis on the quantum annular Khovanov spectrum with an action of a finite cyclic group: we obtain a quantum annular Khovanov spectrum with an action of the infinite cyclic group.

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.

Kervaire conjecture that the weight of the free product of every non-trivial group and the infinite cyclic group is not one is affirmatively confirmed by confirming affirmatively Conjecture Z on the knot exterior introduced by Gonzàlez Acuna and Ramırez as a conjecture equivalent to Kervaire conjecture.

The degree of commutativity of a finite group is the probability that two uniformly and randomly chosen elements commute. This notion extends naturally to finitely generated groups G: the degree of commutativity dcS(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G)$$\\end{document}, with respect to a given finite generating set S, results from considering the fractions of commuting pairs of elements in increasing balls around 1G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1_G$$\\end{document} in the Cayley graph . We focus on restricted wreath products of the form G=H≀⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G = H \\hspace{1.111pt}{\\wr }\\hspace{1.111pt}\\langle \\hspace{1.111pt}t \\rangle $$\\end{document}, where H≠1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \ e 1$$\\end{document} is finitely generated and the top group ⟨t⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\hspace{1.111pt}t \\rangle $$\\end{document} is infinite cyclic. In accordance with a more general conjecture, we show that dcS(G)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{dc}\\,}}_S(G) = 0$$\\end{document} for such groups G, regardless of the choice of S. This extends results of Cox who considered lamplighter groups with respect to certain kinds of generating sets. We also derive a generalisation of Cox’s main auxiliary result: in ‘reasonably large’ homomorphic images of wreath products G as above, the image of the base group has density zero, with respect to certain types of generating sets.

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

This paper aims to treat a study of generators of the cyclic group of higher even, odd, and prime order for composition being multiplication. In fact we developed order of a group, order of element of a group and generators of the cyclic group in real numbers. Also we express cyclic and generators of the group for composition in real numbers. Here we discuss the higher order of groups in different types of order, and generators of the cyclic group which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element a m ∈G in which a n =e = identity element, then find Highest Common Factor i.e. (H.C.F) of m and n . When G is a finite group, every element must have finite order but the converse is false. There are infinite groups where each element has a finite order. There may be more than one generator of a cyclic group. Also every cyclic group is necessarily abelian. But show that every infinite cyclic group contains only two generators. Finally, we find out the generators of the cyclic group of higher even, odd and prime order in different types of the group for composition being multiplication.

Abstract We construct a two-state Mealy automaton A over the three-letter alphabet generating a regular branch group G ⁢ ( A ) {G(A)} , which surjects onto the infinite cyclic group. Some algebraic and geometric properties of the group G ⁢ ( A ) {G(A)} are derived. In particular, this group has a nearly finitary subgroup of index two, is amenable, just non-solvable, has exponential growth, and its action on the corresponding regular rooted tree is self-replicating, contracting, but it does not have the congruence subgroup property. We also derive in detail an ascending finite L-presentation for the group G ⁢ ( A ) {G(A)} .

On an example of Nagarajan

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: [Formula: see text]. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for [Formula: see text] the length of the lower central series of [Formula: see text] is equal to 2, [Formula: see text] or [Formula: see text].

The automorphism group of a K3 surface with Picard number two is either the infinite cyclic group or the infinite dihedral group, if it is infinite. In this paper, we determine some conditions for a K3 surface of Picard number two to have the infinite dihedral automorphism group.

We call a graph k-geodetic, for some [Formula: see text], if it is connected and between any two vertices there are at most k geodesics. It is shown that any hyperbolic group with a k-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centralizer of any infinite order element is an infinite cyclic group. These results were known previously only in the case that [Formula: see text]. A key tool used to develop the theorem is a new graph theoretic result concerning “ladder-like structures” in a k-geodetic graph.

We study locally flat, compact, oriented surfaces in 4-manifolds whose exteriors have infinite cyclic fundamental group. We give algebraic topological criteria for two such surfaces, with the same genus g, to be related by an ambient homeomorphism, and further criteria that imply they are ambiently isotopic. Along the way, we provide a classification of a subset of the topological 4-manifolds with infinite cyclic fundamental group, and we apply our results to rim surgery. 1 Introduction We study locally flat embeddings of compact, orientable surfaces in compact, oriented, simply connected topological 4-manifolds, where the complement of the surface has infinite cyclic fundamental group. Extending the terminology for knotted spheres, we call this group the knot group, so we shall study knotted surfaces with knot group Z, or Z-surfaces. We will present algebraic criteria for pairs of Z-surfaces to be ambiently isotopic. As part of the proof we obtain an algebraic classification of a certain subset of the 4-manifolds with boundary and fundamental group Z, simultaneously generalising work of Freedman and Quinn [15] on the closed case, and of Boyer [5] on simply connected 4-manifolds with nonempty boundary; see Section 1.7. We apply our results to show that in simply connected 4-manifolds, 1-twisted rim surgery on a surface with knot group Z yields a topologically ambiently isotopic surface, extending results of Kim and Ruberman [31] and Juhász, Miller and Zemke [24]; see Section 1.5.

In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients.

A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag - Solitar group (𝐺𝐵𝑆 group). Every group is the fundamental group 𝜋1(A) of a suitable labeled graph A. The paper is the review of recent results that fully describe the centralizer dimension and, in some cases, the universal equivalence of groups in terms of labeled graphs.