Nontrivial Elements Research Articles

Classification of the conjugacy classes of the universal covering group of [formula omitted]

In this note, we classify the conjugacy classes of SL˜2(R), the universal covering group of PSL2(R). For any non-central element α∈SL˜2(R), we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers R+) of its image α¯ in PSL2(R); (ii) Direction type: the sign behavior of the induced self-homeomorphism of R determined by the lifting SL˜2(R)↷R of the action PSL2(R)↷S1; (iii) The functionℓ♯: a conjugacy invariant length function introduced by Mochizuki (2016).

On the Total Dominating Set of ${3}/{2}$-Generated Groups

A subset $S$ of a group $G$ is called a total dominating set of $G$ if for any nontrivial element $x\in G$ there is an element $y\in S$ such that $G =\left< x, y\right> $. Tarski monsters, constructed by Olshanskii, are infinite simple groups, any pair of non-commuting elements of which is a total dominating set. In this paper, we construct an infinite non-cyclic and non-simple group having a total dominating set from two elements. This gives a positive answer to Donoven and Harper's question about the existence of infinite groups (other than Tarski monsters) having a finite total dominating set. In addition, our examples have an infinite uniform spread.

Lower spaces of multiplicative lattices

AbstractWe consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, including lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space; we further explore continuous maps between lower spaces.

A T-duality of non-supersymmetric heterotic strings and an implication for Topological Modular Forms

Motivated by recent developments connecting non-supersymmetric heterotic string theory to the theory of Topological Modular Forms (TMF), we show that the worldsheet theory with central charge (17,32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{2} $$\\end{document}) obtained by fibering the (E8)1 × (E8)1 current algebra over the two N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (0, 1) sigma model on S1 with antiperiodic spin structure (such that the E8 factors are exchanged as we go around the circle), is continuously connected to the (E8)2 theory in the Gaiotto Johnson-Freyd Witten sense of going “up and down the RG trajectories”. Combined with the work of Tachikawa and Yamashita, this furnishes a physical derivation of the fact that the (E8)2 theory corresponds to the unique nontrivial torsion element [(E8)2] of TMF31 with zero mod-2 elliptic genus.

Probabilistic generation of finite almost simple groups

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x∈G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to ∞). This is new even if G is simple. Together with results of Lucchini and Burness–Guralnick–Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.

Residual finiteness growth in virtually abelian groups

A group G is called residually finite if for every non-trivial element g∈G, there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g, for example by using the word norm ‖g‖G if the group G is assumed to be finitely generated. The residual finiteness growth RFG:N→N is then defined as the smallest function such that if ‖g‖G≤r, there exists a morphism φ:G→Q to a finite group Q with |Q|≤RFG(r) and φ(g)≠eQ.Although upper bounds have been established for several classes of groups, exact asymptotics for the function RFG are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals logk for some k∈N, where the value k is given by an explicit expression. As an application, we show that for every m≥1 and every 1≤k≤m, there exists a group G containing a normal abelian subgroup of rank m and with RFG≈logk.

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

Sombor Index and Sombor Polynomial of the Noncommuting Graph Associated to Some Finite Groups

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-central elements of the group and two vertices are adjacent if and only if they do not commute. In this study, a new notion called the Sombor polynomial is introduced. Then, the general formula of the Sombor index and the Sombor polynomial of the noncommuting graph associated to some finite groups are determined by using their definitions and some preliminaries. The groups involved in this research are the dihedral groups, the quasidihedral groups, and the generalized quaternion groups. The results found can help the chemists and biologists to predict the chemical and physical properties of the molecules without involving any laboratory work.

On the consistency of ZF with an elementary embedding from Vλ+2 into Vλ+2

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe [Formula: see text] (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming ZF [Formula: see text] [Formula: see text] [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model [Formula: see text] containing [Formula: see text] and satisfying ZF [Formula: see text] [Formula: see text] [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text] ⊆[Formula: see text], where [Formula: see text] witnesses [Formula: see text] in [Formula: see text]. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text]-DC holds in [Formula: see text], then the model [Formula: see text] will also satisfy [Formula: see text]-DC. We show that ZFC [Formula: see text] “[Formula: see text] is even” [Formula: see text] [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists.

Projective non-commuting graph of a group

Let $G$ be a finite non-abelian group and let $T$ be a transversal of the center of $G$ in $G$. The non-commuting graph of $G$ on a transversal of the center is the graph whose vertices are the non-central elements of $T$ and two vertices $x$ and $y$ are joined by an edge whenever $xy \neq yx$. In this paper, we classify the groups whose non-commuting graph on a transversal of the center is projective.

A note on Wedderburn and Hasse theorems

Wedderburn and Hasse theorems have always fascinated many algebraists as there are just very few versions of their proofs and that are somehow not easy to follow because they contain other results inside. In this note, we aim to look at their proofs again but nowwith more simplified versions that include sufficient details in order to let them be understandable for most readers in the subjects of finite fields and elliptic curves. In the proof of Wedderburn’s theorem, we succeed to show how it is equivalent to define either a stabilizer or centralizer of non-central elements and that helps to consider the equation on the order of group by using interchangeably the index of its subgroup and the order of the orbit. As for Hasse’s theorem, we provide a short proof in just two paragraphs with a recall of the most important results that have been used, and then, we state three of its version with an explanation of how this all leads in the end to the need of an algorithm like the one of René Schoof in 1985.

Note on the limit property of triangular norms1

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f ( x ) = lim n → ∞ x T ( n ) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = T M .

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

Universal adjacency spectrum of the commuting and non-commuting graphs on AC-groups

For a simple undirected graph [Formula: see text] and [Formula: see text], [Formula: see text], the universal adjacency matrix [Formula: see text], where [Formula: see text] is the identity matrix, [Formula: see text] is the all-ones matrix, and [Formula: see text] and [Formula: see text] are the adjacency and degree diagonal matrices of [Formula: see text], respectively. For a finite non-abelian group [Formula: see text] and [Formula: see text], the commuting graph [Formula: see text] is a simple undirected graph with vertex set [Formula: see text] and the adjacency rule is commuting. The non-commuting graph of [Formula: see text] is the complement of [Formula: see text]. An AC-group is a non-abelian group such that the centralizer of every non-central element is abelian. In this paper, we obtain the universal adjacency eigenpairs of the commuting and non-commuting graphs for an arbitrary AC-group [Formula: see text] with [Formula: see text] or [Formula: see text]. Moreover, we determine the eigenpairs of [Formula: see text] when [Formula: see text] is a specific kind of group.

On ∗-reverse derivable maps

Let R be a ring with involution containing a nontrivial symmetric idempotent element e. Let δ : R → R be a mapping such that δ(ab) = δ(b)a∗ + b∗δ(a) for all a, b ∈ R, we call δ a ∗−reverse derivable map on R. In this paper, our aim is to show that under some suitable restrictions imposed on R, every ∗−reverse derivable map of R is additive.

IDEMPOTENT ELEMENTS IN MATRIX RING OF ORDER 2 OVER POLYNOMIAL RING $\mathbb{Z}_{p^2q}[x

An idempotent element in the algebraic structure of a ring is an element that, when multiplied by itself, yields an outcome that remains unchanged and identical to the original element. Any ring with a unity element generally has two idempotent elements, 0 and 1, these particular idempotent elements are commonly referred to as the trivial idempotent elements However, in the case of rings $\mathbb{Z}_n$ and $\mathbb{Z}_n[x]$ it is possible to have non-trivial idempotent elements. In this paper, we will investigate the idempotent elements in the polynomial ring $\mathbb{Z}_{p^2q}[x]$ with $p,q$ different primes. Furthermore, the form and characteristics of non-trivial idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$ will be investigated. The results showed that there are 4 idempotent elements in $\mathbb{Z}_{p^2q}[x]$ and 7 idempotent elements in $M_2(\mathbb{Z}_{p^2q}[x])$.

Products of normal subsets

In this paper we consider which families of finite simple groups G G have the property that for each ϵ > 0 \epsilon > 0 there exists N > 0 N > 0 such that, if | G | ≥ N |G| \ge N and S , T S, T are normal subsets of G G with at least ϵ | G | \epsilon |G| elements each, then every non-trivial element of G G is the product of an element of S S and an element of T T . We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form P S L n ( q ) \mathrm {PSL}_n(q) where q q is fixed and n → ∞ n\to \infty . However, in the case S = T S=T and G G alternating this holds with an explicit bound on N N in terms of ϵ \epsilon . Related problems and applications are also discussed. In particular we show that, if w 1 , w 2 w_1, w_2 are non-trivial words, G G is a finite simple group of Lie type of bounded rank, and for g ∈ G g \in G , P w 1 ( G ) , w 2 ( G ) ( g ) P_{w_1(G),w_2(G)}(g) denotes the probability that g 1 g 2 = g g_1g_2 = g where g i ∈ w i ( G ) g_i \in w_i(G) are chosen uniformly and independently, then, as | G | → ∞ |G| \to \infty , the distribution P w 1 ( G ) , w 2 ( G ) P_{w_1(G),w_2(G)} tends to the uniform distribution on G G with respect to the L ∞ L^{\infty } norm.

Conjugacy classes and conjugate graph of a K-metacyclic group

The relationship between algebraic structures and graphs has become an interesting topic of research nowadays. In this paper, we have considered the conjugate graph related to the conjugacy relation of a group. The vertices of the said graph are the noncentral elements of the group, and two vertices are adjacent if they are conjugate. For this particular study, we focused on the conjugate graph of a K-metacyclic group of order p(p-1). We first determine the conjugacy classes of this group and then obtain its conjugate graph. Various graph properties such as planarity, line graph, complement graph, clique number, dominating number, spectrum, and Laplacian are also studied in this paper.

Complicated generalized torsion elements in Seifert fibered spaces with boundary

In a group, a nontrivial element is called a generalized torsion element if some non-empty finite product of its conjugates is equal to the identity. There are various examples of torsion-free groups which contain generalized torsion elements. We can define the order of a generalized torsion element as the minimum number of its conjugates required to generate the identity. In previous works, three-manifold groups which contain a generalized torsion element of order two are determined. However, there are few previous studies that examine the order of a generalized torsion element bigger than two. In this paper, we focus on Seifert fibered spaces with boundary, including the torus knot exteriors, and construct concretely generalized torsion elements of order [Formula: see text], [Formula: see text], [Formula: see text] and others in their fundamental groups.

Generalization of Randic ́ Index of the Non-commuting Graph for Some Finite Groups

Randić index is one of the classical graph-based molecular structure descriptors in the field of mathematical chemistry. The Randić index of a graph is calculated by summing the reciprocals of the square root of the product of the degrees of two adjacent vertices in the graph. Meanwhile, the non-commuting graph is the graph of vertex set whose vertices are non-central elements and two distinct vertices are joined by an edge if and only if they do not commute. In this paper, the general formula of the Randić index of the non-commuting graph associated to three types of finite groups are presented. The groups involved are the dihedral groups, the generalized quaternion groups, and the quasi-dihedral groups. Some examples of the Randić index of the non-commuting graph related to a certain order of these groups are also given based on the main results.