Non-abelian Group Research Articles

A Brief Survey on the Inverse Galois Problem

Inverse Galois problem (IGP) states whether any finite group is realizable as a Galois group over the field K. It is the question of the structure and representation of the Galois group and also questions its epimorphic images. So, it is called an inverse Galois problem. For K=ℚ (the field of rational number), it is called a classical inverse Galois problem (CIGP). This paper reviews the positive answer to the classical inverse Galois problem (CIGP) for all finite abelian groups and some finite non-abelian solvable groups. We also discuss this problem (CIGP) for some finite non-solvable groups in this paper. This problem still remains to solve, but if we find the true value of the statement ‘All subgroups of order m of the symmetric group (Sm) for all m are realizable as Galois group over ℚ’ then its truth value gives the answer of CIGP. We check this statement for m=1,2,3,4 and 5 in this paper, where we get that this statement is true. If this statement is true, then CIGP has a positive answer. But if this statement is false then CIGP has a negative answer.

Pure subgroups of non-abelian groups

Pure subgroups of non-abelian groups

Deterministic algorithms for the hidden subgroup problem

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~$ \Order(\sqrt{ n}\, )$, where~$n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a~$\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected\/} query complexity that is sensitive to the input, namely~$ \Order(\sqrt{ n/m}\, )$, where~$m$ is the order of the hidden subgroup. In the first version of this article~\cite[Sec.~5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li~\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian\/} groups which solve the problem with~$ \Order(\sqrt{ n/m }\, )$ queries, and find the hidden subgroup with~$ \Order( \sqrt{ n (\log m) / m} + \log m ) $ queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~$\order(\sqrt{ n/m } \,)$, and that of \emph{finding\/} the entire subgroup may also be~$\order(\sqrt{ n/m } \,)$ or even~$\upomega(\sqrt{ n/m } \,) $.}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~$ \Order(\sqrt{ n/m }\, )$ for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~$ \Order(\sqrt{ (n/m) \log (n/m) }\,) $ for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~$\log m$ multiplicative factor increase in the query complexity.

3d mathcal{N} = 2 brane webs and quiver matrices

We discuss 3d brane webs and effective Chern-Simons levels for 3d mathcal{N} = 2 gauge theories. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping positions of D5-branes. We interpret flips as ST-transformations for chiral multiplets. ST-transformations could turn abelian theories into dual theories with mixed Chern-Simons levels that are interpreted as quiver matrices Cij encoding DT-invariants. We notice that each brane web corresponds to a quiver matrix. ST-transformations of holomorphic blocks are discussed to verify results. We also discuss the movement of flavor D5-branes, which leads to double-layer brane webs and manifests fiber-base duality. In the second part, we compute refined vortex partition functions of nonabelian theories with the gauge group U(N) and find corresponding quiver matrices. The computation shows that on Higgs branch nonabelian groups are broken to abelian groups.

Characterizations of Some Groups in Terms of Centralizers

A group G is said to be n-centralizer if its number of element centralizers \(\mid {{\,\mathrm{Cent}\,}}(G)\mid =n\), an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any non-abelian n-centralizer group G, we prove that \(\mid \frac{G}{Z(G)}\mid \le (n-2)^2\), if \(n \le 12\) and \(\mid \frac{G}{Z(G)}\mid \le 2(n-4)^{{log}_2^{(n-4)}}\) otherwise, which improves an earlier result. We prove that if G is an arbitrary non-abelian n-centralizer F-group, then gcd\((n-2, \mid \frac{G}{Z(G)}\mid ) \ne 1\). For a finite F-group G, we show that \(\mid {{\,\mathrm{Cent}\,}}(G)\mid \ge \frac{\mid G \mid }{2}\) iff \(G \cong A_4 \), an extraspecial 2-group or a Frobenius group with abelian kernel and complement of order 2. Among other results, for a finite group G with non-trivial center, it is proved that \(\mid {{\,\mathrm{Cent}\,}}(G)\mid = \frac{\mid G \mid }{2}\) iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.

On the geometry of positive cones in finitely generated groups

We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups. We introduce the Hucha property and the Prieto property for left-orderable groups. We say that a group has the Hucha property if in any left-order the corresponding positive cone is not coarsely connected, and the Prieto property if in any left-order the corresponding positive cone is coarsely connected. We show that all left-orderable free products have the Hucha property, and that the Hucha property is stable under certain free products with amalgamatation over Prieto subgroups. As an application we show that non-abelian limit groups in the sense of Z. Sela (e.g., free groups, fundamental group of hyperbolic surfaces, doubles of free groups and others) and non-abelian finitely generated subgroups of free Q $\mathbb {Q}$ -groups in the sense of G. Baumslag have the Hucha property. In particular, this implies that these groups have empty BNS-invariant Σ 1 $\Sigma ^1$ and that they do not have finitely generated positive cones.

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.

Non-Abelian half-quantum vortices in 3P2 topological superfluids

${}^{3}P_{2}$ superfluids realized in neutron stars are the largest topological quantum matters in our Universe. We establish the existence and stability of non-Abelian half-quantum vortices (HQVs) in ${}^{3}P_{2}$ superfluids with strong magnetic fields. Using a self-consistent microscopic approach, we find that a singly quantized vortex is energetically destabilized into a pair of two non-Abelian HQVs owing to the strongly spin-orbit-coupled pairing. We find a topologically protected Majorana fermion on each HQV, thereby providing two-fold non-Abelian anyons characterized by both Majorana fermions and a non-Abelian first homotopy group.

Primitive quantum gates for dihedral gauge theories

We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$ -- the dihedral group -- serves as an approximation to $U(1)\times\mathbb{Z}_2$ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the nonabelian Fourier transform over $D_N$, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in $n=\log N$. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of $D_4$. The fidelity of all $D_4$ gates was found to exceed $80\%$.

Connectivity of superpower graphs of some non-abelian finite groups

For a finite group [Formula: see text] the superpower graph [Formula: see text] of [Formula: see text] is an undirected simple graph with vertex set [Formula: see text] and two vertices in [Formula: see text] are adjacent in [Formula: see text] if and only if the order of one divides the order of the other in the group [Formula: see text]. In this paper, we give sharp bounds for the vertex connectivity of superpower graphs [Formula: see text] and [Formula: see text] of dihedral group [Formula: see text] and dicyclic group [Formula: see text]

The extension problem in free harmonic analysis

This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of ‘relative energy’ which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szegö parameters.

2-Arc-transitive hexavalent Cayley graphs on nonabelian simple groups

A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency 4 (provided 2-arc-transitivity) by Fang et al. [Europ. J. Combin. 25 (2004), 1107–1116] and Du and Feng [Comm. Algebra 47 (2019), 4565–4574]. This naturally motivates this work for classifying 2-arc-transitive hexavalent Cayley graphs on nonabelian simple groups. It is proved that these graphs are either normal or -arc-transitive Cayley graphs on where n is among 11 specific numbers dividing A specific example is also constructed.

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and Parcet.

On the isomorphism of unitary subgroups of noncommutative group algebras

Let FGFG be the group algebra of a finite pp-group GG over a field FF of characteristic pp. Let *\cd be an involution of the group algebra FGFG which arises form the group basis GG. The upper bound for the number of non-isomorphic *\cd-unitary subgroups is the number of conjugacy classes of the automorphism group GG with all the elements of order two. The upper bound is not always reached in the case when GG is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian pp-group GG whose group algebra FGFG has sharply less number of non-isomorphic *\cd-unitary subgroups than the given upper bound.

An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups

One of the remarkable contributions in the study of symmetric Cayley graphs on nonabelian simple groups is the complete classification of such graphs that are cubic and nonnormal. This naturally motivates the study of cubic (normal and nonnormal) symmetric bi-Cayley graphs on nonabelian simple groups. In this paper, the full automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for a graph being a cubic normal symmetric Cayley or bi-Cayley graph on a nonabelian simple group (one may then find many examples). As an application, we also prove that cubic symmetric Cayley graphs on nonabelian simple groups are stable.

Dihedral Multi-Reference Alignment

We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from a generic distribution, the orbit of a generic signal is uniquely determined from the second moment of the observations. This implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise. This is the first result of this type for multi-reference alignment over a non-abelian group with a non-uniform distribution of group elements. Based on tools from invariant theory and algebraic geometry, we also delineate conditions for unique orbit recovery for multi-reference alignment models over finite groups (namely, when the dihedral group is replaced by a general finite group) when the group elements are drawn from a generic distribution. Finally, we design and study numerically three computational frameworks for estimating the signal based on group synchronization, expectation-maximization, and the method of moments.

Line graph characterization of power graphs of finite nilpotent groups

This paper deals with the classification of groups G such that power graphs and proper power graphs of G are line graphs. In fact, we classify all finite nilpotent groups whose power graphs are line graphs. Also, we categorize all finite nilpotent groups (except non-abelian 2-groups) whose proper power graphs are line graphs. Moreover, we investigate when the proper power graphs of generalized quaternion groups are line graphs. Besides, we derive a condition on the order of the dihedral groups for which the proper power graphs of the dihedral groups are line graphs.

Negative Latin square type partial difference sets in nonabelian groups of order 64

There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64.

The character degree ratio and large abelian subgroups

Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. M. Isaacs defined the character degree ratio of G to be rat(G)=b(G)/m(G). He proved that for nonabelian solvable groups the derived length is bounded by a logarithmic function of the character degree ratio. In a similar spirit, results that restrict the structure of nonsolvable groups in terms of the character degree ratio were later proved by J. P. Cossey, M. Lewis and H. N. Nguyen. In this note we show that, with some rather precisely determined exceptions, G has an abelian subgroup of index bounded by a polynomial function in rat(G). As a consequence we get stronger (in most situations) versions of most of the previously known results on the character degree ratio.

Lattice model constructions for gapless domain walls between topological phases

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.