Solvable Groups Research Articles

Addendum to “A generalization of a result on the sum of element orders of a finite group”

Abstract Let G be a group of order n and H be a subgroup of order m of G. Denote by ψ H (G) the sum of element orders relative to H of G. It is known that if G is nilpotent, then ψ H ( G ) ≤ ψ H m ( C n ) $ \psi_H(G) \leq\psi_{H_m}(C_n) $ , where H m is the unique subgroup of order m of C n . In this note, we show that this inequality does not hold for infinitely many finite solvable groups.

Codegrees and element orders of almost simple groups

G. Qian proposed a conjecture which states that if an element of a finite group G has order a, then there exists an irreducible character of codegree b of G such that a divides b. He showed that the conjecture holds for solvable groups. In this note, we settle the conjecture for almost simple groups.

Conjugacy classes of small size in solvable groups

If G is a finite group and , we say that x lies in a small class if is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra.

Monomial characters of finite solvable groups

We give new evidence to the fact that the structure of a solvable group can be controlled by irreducible monomial characters. In particular, we inspect the role of monomial characters in the Isaacs–Navarro–Wolf conjecture and in Gluck’s conjecture.

ON LARGE ORBITS OF FINITE SOLVABLE GROUPS ON CHARACTERS

AbstractWe prove that if a solvable group A acts coprimely on a solvable group G, then A has a relatively ‘large’ orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This improves an earlier result of Keller and Yang [‘Orbits of finite solvable groups on characters’, Israel J. Math.199 (2014), 933–940].

Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question

Abstract The Gruenberg–Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well-studied class of rational groups, and has received attention recently. In the first part of this paper, we give a complete classification of the Gruenberg–Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg–Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg–Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg–Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The prime graph question asks whether the Gruenberg–Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the first Zassenhaus conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the prime graph question for integral group rings for finite rational groups and most finite cut groups

Classification of non-solvable groups whose power graph is a cograph

Abstract In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.

FRÖLICHER SPECTRAL SEQUENCE AND HODGE STRUCTURES ON THE COHOMOLOGY OF COMPLEX PARALLELISABLE MANIFOLDS

For complex parallelisable manifolds Γ\\G, with G a solvable or semisimple complex Lie group, the Frölicher spectral sequence degenerates at the second page. In the solvable case, the de Rham cohomology carries a pure Hodge structure. In contrast, in the semisimple case, purity depends on the lattice, but there is always a direct summand of the de Rham cohomology which does carry a pure Hodge structure and is independent of the lattice.

Critical metrics for quadratic curvature functionals on some solvmanifolds

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G_{mathcal S}=H rtimes exp ({mathbb {R}}S) of the Heisenberg Lie group H, for any mathcal S in {mathfrak {s}}{mathfrak {p}}(1,mathbb R), equipped with a family g_a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g_a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals.

The average character degree of finite groups and Gluck’s conjecture

Abstract We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of acd ⁡ ( G ) \operatorname{acd}(G) , the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of acd ⁡ ( G ) \operatorname{acd}(G) , but we show that, in certain cases, it is bounded in terms of the degrees of the irreducible characters of 𝐺 that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck’s conjecture.

Compositions and parities of complete mappings and of orthomorphisms

We determine the permutation groups Pcomp(Fq),Porth(Fq)≤Sym(Fq) generated by the complete mappings, respectively the orthomorphisms, of the finite field Fq – both are equal to Sym(Fq) unless q∈{2,3,4,5,8}. More generally, denote by Pcomp(G), respectively Porth(G), the subgroup of Sym(G) generated by the complete mappings, respectively the orthomorphisms, of the group G. Using recent results of Eberhard-Manners-Mrazović and Müyesser-Pokrovskiy, we show that for each large enough finite group G that has a complete mapping (i.e., whose Sylow 2-subgroups are trivial or noncyclic), Pcomp(G)=Sym(G) and Porth(G)≥Alt(G). We also prove that Porth(G)=Sym(G) for every large enough finite solvable group G that has a complete mapping. Proving these results requires us to study the parities of complete mappings and of orthomorphisms. Some connections with known results in cryptography and with parity types of Latin squares are also discussed.

Almost co-Kähler manifolds and [formula omitted]-quasi-Einstein solitons

The present paper aims to investigate (m,ρ)-quasi-Einstein metrics on almost co-Kähler manifolds M. It is proven that if a (κ,μ)-almost co-Kähler manifold with κ<0 is (m,ρ)-quasi-Einstein manifold, then M represents a N(κ)-almost co-Kähler manifold and the manifold is locally isomorphic to a solvable non-nilpotent Lie group. Next, we study the three dimensional case and get the above mentioned result along with the manifold M3 becoming an η-Einstein manifold. We also show that there does not exist (m,ρ)-quasi-Einstein structure on a compact (κ,μ)-almost co-Kähler manifold of dimension greater than three with κ<0. Further, we prove that an almost co-Kähler manifold satisfying η-Einstein condition with constant coefficients reduces to a K-almost co-Kähler manifold, provided ma1≠(2n−1)b1 and m≠1. We also characterize perfect fluid spacetime whose Lorentzian metric is equipped with (m,ρ)-quasi Einstein solitons and acquired that the perfect fluid spacetime has vanishing vorticity, or it represents dark energy era under certain restriction on the potential function. Finally, we construct an example of an almost co-Kähler manifold with (m,ρ)-quasi-Einstein solitons.

Polynomially growing harmonic functions on connected groups

We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties. Our main result shows that (for sufficiently “nice” random walk measures) a connected, compactly generated, locally compact group has polynomial volume growth if and only if the space of linear growth harmonic functions has finite dimension. This characterization is interesting in light of the fact that Gromov’s theorem regarding finitely generated groups of polynomial growth does not have an analog in the connected case. That is, there are examples of connected groups of polynomial growth that are not nilpotent by compact. Also, the analogous result for the discrete case has only been established for solvable groups and is still open for general finitely generated groups.

Extraction Algorithm of HOM–LIE Algebras Based on Solvable and Nilpotent Groups

Hom–Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom–Lie algebras are studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well‐developed. We establish a theory of the solvable and nilpotent Hom–Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research.Dedicated to Al Farouk School &amp; Kinder garten-Irbid-Jordan

Problems on characters: solvable groups

Problems on characters: solvable groups

Almost Kähler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost K?hler structures on that group. The only K?hler structure is the standard structure of the complex hyperbolic space and all others are strictly almost K?hler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K?hler one.

On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows

We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^{n}$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

Finite Solvable Groups in Which the $\sigma$-Quasinormality of Subgroups is a Transitive Relation

Пусть $\sigma=\{\sigma_{i} \mid i\in I\}$ - некоторое разбиение множества всех простых чисел и $G$ - конечная группа. Группа $G$ называется: $\sigma$-примарной, если $G$ является $\sigma_{i}$-группой для некоторого $i\in I$; $\sigma$-полной, если $G$ имеет холлову $\sigma_{i}$-подгруппу для всех $i\in I$. Говорят, что подгруппа $A$ группы $G$: (i) $\sigma$-субнормальна в $G$, если существует цепь подгрупп $A=A_{0} \leqslant A_{1} \leqslant \dotsb \leqslant A_{n}=G$ такая, что либо $A_{i-1} \trianglelefteq A_{i}$, либо $A_{i}/(A_{i-1})_{A_{i}}$ является ${\sigma}$-примарной для всех $i=1, …, n$; (ii) модулярной в $G$, если выполняются следующие условия: (1) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ для всех $X \leqslant G, Z \leqslant G$ таких, что $X \leqslant Z$, и (2) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ для всех $Y \leqslant G, Z \leqslant G$ таких, что $A \leqslant Z$; (iii) $\sigma$-квазинормальной в $G$, если $A$ $\sigma$-субнормальна и модулярна в $G$. В работе получено описание конечных разрешимых групп c транзитивным отношением $\sigma$-квазинормальности подгрупп. Обобщаются некоторые известные результаты. Библиография: 16 названий.

Moduli spaces of morphisms into solvable algebraic groups

We construct (in significant generality) moduli spaces representing the functor of morphisms from a scheme into a solvable algebraic group.

Measurement as a Shortcut to Long-Range Entangled Quantum Matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.