Solvable Groups Research Articles

On semiabelian groups

Abstract Semiabelian groups are a family of finite solvable groups that have played an important role in the inverse Galois problem and the minimal ramification problem. In this paper, we establish several sufficient conditions for a solvable group to be semiabelian mainly by using its Sylow subgroups. We also present a family of non-semiabelian 2-groups.

Solvable groups with four supercharacter theories

Solvable groups with four supercharacter theories

Kernels of minimal characters of solvable groups

Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

Bott–Chern Formality and Massey Products on Strong Kähler with Torsion and Kähler Solvmanifolds

We study the interplay between geometrically-Bott–Chern-formal metrics and SKT metrics. We prove that a 6-dimensional nilmanifold endowed with a invariant complex structure admits an SKT metric if and only if it is geometrically-Bott–Chern-formal. We also provide some partial results in higher dimensions for nilmanifolds endowed with a class of suitable complex structures. Furthermore, we prove that any Kähler solvmanifold is geometrically formal. Finally, we explicitly construct lattices for a complex solvable Lie group in the list of Nakamura (J Differ Geom 10:85–112, 1975) on which we provide a non vanishing quadruple ABC-Massey product.

A family of graphs that cannot occur as character degree graphs of solvable groups

A family of graphs that cannot occur as character degree graphs of solvable groups

Improved algebraic fibrings

We show that a virtually residually finite rationally solvable (RFRS) group $G$ of type $\mathtt {FP}_n(\mathbb {Q})$ virtually algebraically fibres with kernel of type $\mathtt {FP}_n(\mathbb {Q})$ if and only if the first $n$$\ell ^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$ are virtually Abelian. It then follows that if $G$ is a virtually RFRS group of type $\mathtt {FP}(\mathbb {Q})$ such that $\mathbb {Z} G$ is Noetherian, then $G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.

A geometric Elliott invariant and noncommutative rigidity of mapping tori

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C⁎-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C⁎-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

Lorentzian distance on the Lobachevsky plane *

Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.

On Gluck’s conjecture for wreath product type groups

Abstract A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

A Note on the Codegree of Finite Groups

Let χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document} be an irreducible character of a group G, and Sc(G)=∑χ∈Irr(G)cod(χ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_c(G)=\\sum _{\\chi \\in \ extrm{Irr}(G)}\ extrm{cod}(\\chi )$$\\end{document} be the sum of the codegrees of the irreducible characters of G. Write fcod(G)=Sc(G)|G|.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{fcod} (G)=\\frac{S_c(G)}{|G|}.$$\\end{document} We aim to explore the structure of finite groups in terms of fcod(G).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{fcod} (G).$$\\end{document} On the other hand, we determine the lower bound of Sc(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_c(G)$$\\end{document} for nonsolvable groups and prove that if G is nonsolvable, then Sc(G)⩾Sc(A5)=68,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_c(G)\\geqslant S_c(A_5)=68,$$\\end{document} with equality if and only if G≅A5.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\cong A_5.$$\\end{document} Additionally, we show that there is a solvable group so that it has the codegree sum as A5.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_5.$$\\end{document}

A height gap in $\mathrm{GL}_{d}(\overline{\mathbb{Q}})$ and almost laws

E. Breuillard showed that finite subsets F\hspace{-.75pt} of matrices in \mathrm{GL}_{d\hspace{-.75pt}}(\hspace{-.75pt}\overline{\mathbb{Q}}\hspace{-.75pt}) generating non-virtually solvable groups have normalized height \hat{h}(F)\ge\varepsilon_d , for some positive \varepsilon_{d}>0 . The normalized height \hat{h}(F) is a measure of the arithmetic size of F and this result can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.

The prime-power hypothesis on the codegrees of irreducible characters

For a character χ of a finite group G, the number cod ( χ ) = [ G : Ker χ ] χ ( 1 ) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod ( χ ) ≠ cod ( ϕ ) , the greatest common divisor of cod ( χ ) and cod ( ϕ ) is a prime-power, then either G is solvable or G / Sol ( G ) is isomorphic to one of the groups Alt 5 or PSL 2 ( 8 ) , where Sol ( G ) is the solvable radical of the group G.

A new lower bound for the number of conjugacy classes

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G G is a finite group, p p is a prime dividing the group order, and k ( G ) k(G) is the number of conjugacy classes of G G , then k ( G ) ≥ 2 p − 1 k(G)\geq 2\sqrt {p-1} , and they proved this conjecture for solvable G G and showed that it is sharp for those primes p p for which p − 1 \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p p , and we prove it for solvable groups, and when p p is large, also for arbitrary groups.

Finite Groups Whose Maximal Subgroups are 2-Nilpotent or Normal

We describe the structure of those finite groups whose maximal subgroups are either 2-nilpotent or normal. Among other properties, we prove that if such a group G does not have any non-trivial quotient that is a 2-group, then G is solvable. Also, if G is a solvable group satisfying the above conditions, then the 2-length of G is less than or equal to 2. If, on the contrary, G is not solvable, then G has exactly one non-abelian principal factor and the unique simple group involved is one of the groups PSL2(p2a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PSL}_2(p^{2^a})$$\\end{document}, where p is an odd prime and a≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\ge 1$$\\end{document}, or p is a prime satisfying p≡±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\equiv \\pm 1$$\\end{document}(mod8)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extrm{mod}~ 8)$$\\end{document} and a=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a=0$$\\end{document}.

On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

On the regular representation of solvable Lie groups with open coadjoint quasi-orbits

Reductive covers of klt varieties

In this article, we study G -covers of klt varieties, where G is a reductive group. First, we exhibit an example of a klt singularity admitting a \mathbb{P}{\mathrm{GL}}_{n}(\mathbb{K}) -cover that is not of klt type. Then, we restrict ourselves to G -quasi-torsors, a special class of G -covers that behave like G -torsors outside closed subsets of codimension two. Given a G -quasi-torsor X\rightarrow Y , where G is a finite extension of a torus \mathbb{T} , we show that X is of klt type if and only if Y is of klt type. We prove a structural theorem for \mathbb{T} -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of \mathbb{T} -quasi-torsors over a variety with klt type singularities is eventually a sequence of \mathbb{T} -torsors. This is the torus version of a result due to Greb–Kebekus–Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and \mathbb{T} -quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.

On the number of conjugacy classes of a finite solvable group

On the number of conjugacy classes of a finite solvable group

On an IC-property of subgroups of a finite group

Let G be a finite group and H a subgroup of G. We say that H has an IC-property if the intersection of H and [ H , G ] satisfy some conditions. In this paper, we investigate the structure of a finite group G under the assumption that some maximal subgroups, 2-maximal subgroups, 3-maximal subgroups of G have the ICp -property and obtain some new characterizations of p-solvable and solvable groups.

Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G×H symmetry gives rise to anomalous LRE upon measuring G-charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024

Upper bounds for the diameter of a direct power of non-abelian solvable groups

Upper bounds for the diameter of a direct power of non-abelian solvable groups