Non-cyclic Group Research Articles

Frattini subgroups of hyperbolic-like groups

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group G admitting a general type action on a hyperbolic space S, we show that the induced action of the Frattini subgroup Φ(G) on S has bounded orbits. This implies that Φ(G) is “small” compared to G; in particular, |G:Φ(G)|=∞. In contrast, for any finitely generated non-cyclic group Q with Φ(Q)={1}, we construct an infinite lacunary hyperbolic group L such that L/Φ(L)≅Q; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.

Non-Abelian extensions of degree p3 and p4 in characteristic p > 2

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order p3 or the ten non-Abelian groups of order p4, p an odd prime, over a field of characteristic p.

Lee metrics on groups

In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and not to have interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $ G $ is a torsion-free group or a finite group of odd order, we prove that $ G $ has a Lee metric if and only if $ G $ is cyclic. Also, if $ G $ is a group admitting Lee metrics then $ G \times \mathbb{Z}_2^k $ always has Lee metrics for every $ k \in \mathbb{N} $. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $ \le 31 $ indicating which of them have (or have not) Lee metrics and why (not).

An exact upper bound for the sum of powers of element orders in non-cyclic finite groups

For a finite group G, let ψ(G) denote the sum of element orders of G. This function was introduced by H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs in 2009 and they proved that for any finite group G of order n, ψ(G) is maximum if and only if G≃Zn where Zn denotes the cyclic group of order n. Furthermore, M. Herzog, P. Longobardi, and M. Maj in 2018 proved that if G is non-cyclic, ψ(G)≤711ψ(Zn). S. M. Jafarian Amiri and M. Amiri in 2014 introduced the function ψk(G) which is defined as the sum of the k-th powers of element orders of G and they showed that for every positive integer k, ψk(G) is also maximum if and only if G is cyclic.In this paper, we have been able to prove that if G is a non-cyclic group of order n, then ψk(G)≤1+3.2k1+2.4k+2kψk(Zn). Setting k=1 in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for ψk(G) is also obtained for finite abelian p-groups G, using which one can explicitly find out the exact value of ψk(G) for finite abelian groups G.

Degree and distance based topological descriptors of power graphs of finite non-abelian groups

A topological descriptor is a numerical value derived from the molecular structure that encapsulates the most important structural characteristics of the molecule under consideration. Fundamentally, it involves assigning an algebraic value to the composition of chemicals while developing a relationship between this value and several physical properties, like biological activity, and chemical reactivity. This article examines multiple kinds of degree and eccentricity-based topological indices for power graphs of various finite groups. We calculate the Wiener index and its reciprocal, atom-bond connectivity index and its fourth version, the Schultz index, the geometric–arithmetic and harmonic indices, and finally determine the general Randić and Harary indices of power graphs of finite cyclic and non-cyclic groups of order pq, dihedral, and generalized quaternion groups, where p,q(q≥p) are distinct primes.

Upper bounds for the product of element orders of finite groups

Let G be a finite group of order n, and denote by rho (G) the product of element orders of G. The aim of this work is to provide some upper bounds for rho (G) depending only on n and on its least prime divisor, when G belongs to some classes of non-cyclic groups.

Hosoya Polynomials of Power Graphs of Certain Finite Groups.

Assume that is a finite group. The power graph of is a graph in which is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order and , where and are prime numbers.

Intuitionistic Level Subgroups in the Dihedral Group D3

A well known result in fuzzy group theory states that “level subgroups of any fuzzy subgroup of a finite group form a chain”. We check the validity of this statement in the intuitionistic fuzzy perspective. We do this using Dihedral Group $D_3$, which is a non-cyclic group. We prove that $D_3$ has 100 distinct intuitionistic fuzzy subgroups (IFSGs) upto isomorphism. The intuitionistic level subgroups (ILSGs) of exactly 76 among them make chains, and hence it can be concluded that the result is not true in the intuitionistic fuzzy perspective. We also enlist all the 100 distinct intuitionistic fuzzy subgroups of $D_3$ upto isomorphism.

Quantifying local embeddings into finite groups

We study a function LΓ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group Γ. We compute this LEF growth function in some examples, including certain wreath products. We compare LEF growth with the analogous quantitative version of residual finiteness, and exhibit a family of finitely generated residually finite groups which nevertheless admit many more local embeddings into finite groups than they do finite quotients. Along the way, we give a new proof that B.H. Neumann's continuous family of 2-generated groups contains no finitely presented group, a result originally due to Baumslag and Miller. We compare LΓ with quantitative versions of soficity and other metric approximation properties of groups. Finally, we show that there exists a “universal” function which is an upper bound on the LEF growth of any group on a given number of generators, and that (for non-cyclic groups) any such function is non-computable.

On a conjecture of the small Davenport constant for finite groups

Let G be a multiplicatively written finite group. We denote by d(G) the small Davenport constant of G, that is, the maximal integer ℓ such that there is a sequence of length ℓ over G which has no non-trivial product-one subsequence. In 2014, Gao, Li, and Peng conjectured that d(G)≤|G|/p+p−2 for any finite non-cyclic group G, where p is the smallest prime divisor of |G|. In this paper, we confirm that this conjecture is true.

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.

Finite non-cyclic nilpotent group whose number of subgroups is minimal

Finite non-cyclic nilpotent group whose number of subgroups is minimal

Comparing upper arm and trunk kinematics between manufacturing workers performing predominantly cyclic and non-cyclic work tasks

Musculoskeletal disorders (MSDs) are common among manufacturing workers. Exposure to non-neutral postures and high movement speeds associated with MSDs among manufacturing workers may depend on the extent of the variability in the work tasks performed (i.e., predominantly “cyclic” versus “non-cyclic” work). The objectives of this study were to (i) compare mean levels of full-shift exposure summary metrics based on both posture and movement speed between manufacturing workers performing predominantly cyclic (n = 18) and non-cyclic (n = 17) tasks, and (ii) explore patterns of between- and within-worker exposure variance and between-minute (within-shift) exposure level and variation within each group. Inertial sensors were used to measure exposures for up to 15 full shifts per participant. Results indicated (i) substantially higher upper arm and trunk movement speeds among workers performing predominantly cyclic tasks relative to workers performing non-cyclic tasks despite similar postures, and (ii) greater exposure variability both between and within workers in the non-cyclic group.

Minimal cut-sets in the power graphs of certain finite non-cyclic groups

We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.

The second maximal groups with respect to the sum of element orders

Denote by G a finite group and let ψ ( G ) denote the sum of element orders in G . In 2009, H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs proved that if | G | = n and G is non-cyclic, then ψ ( G ) < ψ ( C n ) , where C n denotes the cyclic group of order n . In 2018 we proved that if G is non-cyclic group of order n , then ψ ( G ) ≤ 7 11 ψ ( C n ) and equality holds if n = 4 k with ( k , 2 ) = 1 and G = ( C 2 × C 2 ) × C k . In this paper we proved that equality holds if and only if n and G are as indicated above. Moreover we proved the following generalization of this result: Theorem 4 . Let G be a non-cyclic group of order n , with q being the least prime divisor of n . Then ψ ( G ) ≤ ( ( q 2 − 1 ) q + 1 ) ( q + 1 ) q 5 + 1 ψ ( C n ) , with equality if and only if n = q 2 k with ( k , q ) = 1 and G = ( C q × C q ) × C k . Notice that if q = 2 , then ( ( q 2 − 1 ) q + 1 ) ( q + 1 ) q 5 + 1 = 7 11 .

Finite groups, 2-generation and the uniform domination number

Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1,…,xk, there exists y G ∈ G such that G = 〈xi,y〉 for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) ⩾ 2 for every non-abelian finite simple group G. For any group with u(G) ⩾ 1, we define the uniform domination number γu(G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈x, y〉 (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu(G) for all simple groups G.In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2-generation of soluble and symmetric groups, and we present several open problems.

Normal Automorphisms of Free Groups of Infinitely Based Varieties

In the paper, automorphisms are studied for free groups of varieties given by a family of identities in the well-known infinite independent system of identities involving two variables that was constructed by S. I. Adian to solve the finite basis problem in group theory. It is proved that every normal automorphism (i.e., an automorphism that stabilizes any normal subgroup) of noncyclic free groups of these varieties is an inner automorphism.

Finite groups whose noncyclic graphs have positive genus

For a finite noncyclic group G, let $${\rm Cyc} (G)$$ be a set of elements a of G such that $$\langle a, b\rangle$$ is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set $$G\setminus {\rm Cyc} (G)$$ , having an edge between two distinct vertices x and y if $$\langle x, y\rangle$$ is not cyclic. In this paper, we show that, for a fixed nonnegative integer k, there are at most finitely many finite noncyclic groups whose noncyclic graphs have (non)orientable genus k. We also classify the finite noncyclic groups whose noncyclic graphs have (non)orientable genus 1, 2, 3 and 4, respectively.

On the Probability That an Automorphism Fixes a Group Element

Let G be a finite group. Denote by P(G) the probability that two elements of G, selected randomly and with replacement, commute, and by PA(G) the probability that a randomly chosen automorphism of G fixes a randomly chosen element of G. It is known that for any nonabelian group. We show that for any noncyclic group, and then prove that this bound is not sharp. We also prove that if and only if , where denotes the group of all conjugacy class-preserving automorphisms of G. This gives another perspective on a question of Avinoam Mann.

Finite groups with star-free noncyclic graphs

Abstract For a finite noncyclic group G, let Cyc(G) be the set of elements a of G such that 〈a, b〉 is cyclic for each b of G. The noncyclic graph of G is a graph with the vertex set G ∖ Cyc(G), having an edge between two distinct vertices x and y if 〈x, y〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are K1,n-free, where K1,n is a star and 3 ≤ n ≤ 6.