Solvable Groups Research Articles

Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph

The Gruenberg–Kegel graph Γ(G) associated with a finite group G is an undirected graph without loops and multiple edges whose vertices are the prime divisors of |G| and in which vertices p and q are adjacent in Γ(G) if and only if G contains an element of order pq. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg–Kegel graph. However, our main aim is to prove several new results. Among them are the following.•There are infinitely many finite groups with the same Gruenberg–Kegel graph as the Gruenberg–Kegel of a finite group G if and only if there is a finite group H with non-trivial solvable radical such that Γ(G)=Γ(H).•There is a function F on the natural numbers with the property that if a finite n-vertex graph whose vertices are labeled by pairwise distinct primes is the Gruenberg–Kegel graph of more than F(n) finite groups, then it is the Gruenberg–Kegel graph of infinitely many finite groups. (The function we give satisfies F(n)=O(n7), but this is not best possible.)•If a finite graph Γ whose vertices are labeled by pairwise distinct primes is the Gruenberg–Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, Γ has at least three pairwise non-adjacent vertices, and each vertex is non-adjacent to at least one other vertex, in particular, 2 is non-adjacent to at least one odd vertex.•Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg–Kegel graph.•The groups G22(27) and E8(2) are uniquely determined by the isomorphism types of their Gruenberg–Kegel graphs. In addition, we consider groups whose Gruenberg–Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name EPPO groups; completing this line of research, we give a complete list of such groups.

Solvable hypergroups and a generalization of Hall's theorems on finite solvable groups to association schemes

Within the class of hypergroups (in the sense of [8]) we consider a specific class of hypergroups the members of which we call solvable. Our definition of solvable hypergroups generalizes the notion of a solvable group. We investigate solvable hypergroups and utilize our results to generalize Philip Hall's celebrated theorems on finite solvable groups to the theory of association schemes.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Topological Loops with Decomposable Solvable Multiplication Groups

In this paper we deal with the class mathcal {C} of decomposable solvable Lie groups having dimension six. We determine those Lie groups in mathcal {C} and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

Field of values of cut groups and k-rational groups

Motivated by a question of A. Bächle, we prove that if the field of values of any irreducible character of a finite group G is imaginary quadratic or rational, then the field generated by the character table Q(G)/Q is an extension of degree bounded in terms of the largest alternating group that appears as a composition factor of G. In order to prove this result, we extend a theorem of J. Tent on quadratic rational solvable groups to nonsolvable groups.

Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces

AbstractWe consider finite-dimensional Hopf algebras $u$ that admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated with functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$.

C-minimal topological groups

Abstract In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Huppert's conjecture for character codegrees

AbstractHuppert's ρ‐σ conjecture is one of the main open problems on character degrees of finite groups. A number of ρ‐σ problems have been studied. For instance, T. Keller and J. Zhang considered the ρ‐σ problem for element orders in the 1990s. Recently, a lot of research is being done on character codegrees. Y. Yang and G. Qian studied the ρ‐σ problem for character codegrees in 2017. In this note, we obtain a sharp bound for groups with trivial solvable radical. As a consequence, we improve the general bound of Yang and Qian. For solvable groups, we notice that the ρ‐σ problem for character codegrees is very closely related to the ρ‐σ problem for element orders. In particular, we give a partial negative answer to a speculation by Yang and Qian on the exact bound for the ρ‐σ problem for character codegrees.

Regular orbits of finite primitive solvable groups, III

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on a finite vector space V, and G is not metacyclic. Then G always has a regular orbit on V except for a few “small” cases.

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

AbstractLet G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

An Interpolation From Sol to Hyperbolic Space

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at . These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

On a result of nilpotent subgroups of solvable groups

‎Heineken [‎H‎. ‎Heineken‎, ‎Nilpotent subgroups of finite soluble groups‎, Arch‎. ‎Math.(Basel)‎, ‎ 56 no‎. ‎5 (1991) 417--423‎.] studied the order of the nilpotent subgroups of the largest order of a solvable group‎. ‎We point out an error‎, ‎and thus refute the proof of the main result of [‎H‎. ‎Heineken‎, ‎Nilpotent subgroups of finite soluble groups‎, Arch‎. ‎Math.(Basel)}‎, 56 no‎. ‎5 (1991) 417--423‎.].

The perfect groups of order up to two million

We enumerate the 15768 15768 perfect groups of order up to 2 ⋅ 10 6 2\cdot 10^6 , up to isomorphism, thus also completing the missing cases reported by Holt and Plesken [Perfect groups, The Clarendon Press, Oxford University Press, New York, 1989]. The work supplements the by now well-understood computer classifications of solvable groups, illustrating scope and feasibility of the enumeration process for nonsolvable groups.

A new flag-transitive affine plane of order bm$2^9$

Finite flag-transitive affine planes have received much attention during the past fifty years because of their connections with other combinatorial objects such as spreads, planar functions, semifields and linearized polynomials. In 1964, Foulser completely determined the automorphism groups of finite flag-transitive affine planes. If a flag-transitive affine plane has a solvable automorphism group, then the affine plane is called solvable. The non-solvable flag-transitive affine planes have been completely classified in the 1990s. But the complete classification for the solvable case seems far out of reach. All known solvable flag-transitive affine planes can be classified into two types: $\\mathcal{C}$-planes and $\\mathcal{H}$-planes, where $\\mathcal{H}$-planes only occur in the odd characteristic case. In this paper, we construct the first flag-transitive affine plane of order $2^9$ over its kernel $\\bF_{2^3}$, which is not of type $\\mathcal{C}$ and the largest Singer subgroup of the translation complement has order $(2^3-1)(2^9+1)/9$.

The number of set-orbits of a solvable permutation group

Abstract A permutation group G acting on a set Ω induces a permutation action on the power set 𝒫 ⁢ ( Ω ) {\mathscr{P}(\Omega)} (the set of all subsets of Ω). Let G be a finite permutation group of degree n, and let s ⁢ ( G ) {s(G)} denote the number of orbits of G on 𝒫 ⁢ ( Ω ) {\mathscr{P}(\Omega)} . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | {\log_{2}s(G)/{\log_{2}\lvert G\rvert}} over all solvable groups G. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ {p^{\prime}} -degree irreducible representations of a solvable group.

On the Exponent of the Weak Commutativity Group $$\chi (G)$$

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. The group $\chi(G)$ is an extension of $D(G) = [G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\exp(D(G))$ divides $\exp(G)^{d}$ if $|G|$ is odd and $\exp(D(G))$ divides $2^{d-1}\cdot \exp(G)^{d}$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\exp(D(G))$ divides $\exp(G)$. Moreover, if $G$ is a finite $p$-group of class $c\geq 2$, then $\exp(D(G))$ divides $\exp(G)^{\lceil \log_{p-1}(c+1)\rceil}$ ($p\geq 3$) and $\exp(D(G))$ divides $2^{\lfloor \log_2(c)\rfloor} \cdot \exp(G)^{\lfloor \log_2(c)\rfloor+1}$ ($p=2$).

Embedding theorems for solvable groups

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group G G lying in a variety M {\mathcal M} can be embedded in a 4 4 -generated group H ∈ M A H \in {\mathcal M}{\mathcal A} ( A {\mathcal A} means the variety of abelian groups). If G G is a finite group, then H H can also be found as a finite group. It follows, that any finitely generated (finite) solvable group G G of the derived length l l can be embedded in a 4 4 -generated (finite) solvable group H H of length l + 1 l+1 . Thus, we answer the question of V. H. Mikaelian and A. Yu. Olshanskii. It is also shown that any countable group G ∈ M G\in {\mathcal M} , such that the abelianization G a b G_{ab} is a free abelian group, is embeddable in a 2 2 -generated group H ∈ M A H\in {\mathcal M}{\mathcal A} .

PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

AbstractWe consider G, a linear algebraic group defined over $\Bbbk $ , an algebraically closed field (ACF). By considering $\Bbbk $ as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space $S^\mu _G(\Bbbk )$ consisting of $\mu $ -types on G. We show that for each $p_\mu \in S^\mu _G(\Bbbk )$ , $\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$ is a solvable infinite algebraic group when $p_\mu $ is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of $\mathrm {Stab}\left (p_\mu \right )$ in terms of the dimension of p.

On Huppert's rho-sigma conjecture

For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a “strengthened” form of the conjecture in the general case, asking whether |ρ(G)|≤2σ(G)+1 is true for every finite group G. In this paper we study the strengthened ρ-σ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided σ(G)≤5, but it is false in general if σ(G)≥6. Instead, we establish that |ρ(G)|≤3σ(G)−4 holds for every finite group with a trivial Fitting subgroup and with σ(G)≥6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have |ρ(G)|≤3σ(G) whenever G belongs to one particular class including all the finite solvable groups, and we improve the up-to-date best bound obtained in [18] for the general case.