Probability that the commutator of two group elements is equal to a given element

For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g and [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number and dominating number. We prove that if G, H are isoclinic groups with |Z (G)|=|Z (H)|, then their associated graphs are isomorphic.

Expanders are highly connected sparse graphs of great interest in computer science, in areas ranging from parallel computation to complexity theory, from cryptography to coding theory, and, most recently, computational group theory (see e.g. [AKS, G+, LP, SS, V, WZ].) The explicit constructions of expander graphs [M1, M2, LPS] (see also [GG, Lu]) use deep mathematical tools to construct families of Cayley graphs of finite groups. The Independence Problem [LW], is whether being an expander family is a property of the groups alone, independent of the choice of generators. A counterexample to the general the problem was recently obtained in [ALW], by using a new combinatorial construction of expanders [RVW]. The problem remains for all classical series of finite simple groups (An, PSL(2,p), etc.)Let G be a finite group generated by at most d elements. The product replacement graph Γk(G) is defined to be a graph, with vertices corresponding to generating k-tuples in G, and edges corresponding to Nielsen transformations. Most recently, graphs Γk(G) became a subject of an intense investigation, prompted by the study of a commonly used 'practical' product replacement algorithm for generating random elements in finite groups, designed Leedham-Green and Soicher [LG]. This algorithm, based on the random walk on graphs Γk(G), showed a remarkable performance, as reported in [C+]. It was suggested in [LP], and proved in several special cases, that the product replacement graphs Γk(G) are expanders, for a fixed k, when |G| → ∞The main result of this paper is Theorem 2, establishing the connection between the expansion coefficient of the product replacement graph Γk(G) and the minimal expansion coefficient of a Cayley graph of group G with k generators. One can think of our result as of an additional evidence in favor of the speculation in [LP]. On the other hand, it gives an algorithmic motivation for study of the Independence Problem, in the aftermath of [ALW].In a special case (see Theorem 1 below), we show that if one assumes that all Cayley graphs with at most four generators in PSL (2, p) have a universal lower bound on expansion, then the product replacement graphs Γk(PSL(2,p)) form an expander family, when k ≥ 8 is fixed, and p → ∞.The proof of the Theorems consists of two parts: probability on groups and graph theoretic. The latter is based on a new combinatorial result of independent interest (Lemmas 4, 5), which generalizes a classical lemma that the direct product of expanders is an expander (see e.g. [Ch]). We prove that if a graph Γ on a direct product of sets satisfies two conditions:1) all 'row' subgraphs are expanders,2) at least (1-e) proportion of 'column' graphs are expanders,then graph &Gamma is also an expander. We use a novel combinatorial technique based on graph decomposition, as opposed to path arguments used in previously in [CG, DS2, DS3, P4]. It is easy to see that such a technique can never prove that a certain family of graphs is an expander family (cf. [P3]).

Strongly generating elements in finite and profinite groups

A blackbox secret sharing (BBSS) scheme works in exactly the same way for all finite Abelian groups G; it can be instantiated for any such group G and only black-box access to its group operations and to random group elements is required. A secret is a single group element and each of the n players’ shares is a vector of such elements. Share-computation and secret-reconstruction is by integer linear combinations. These do not depend on G, and neither do the privacy and reconstruction parameters t, r. This classical, fundamental primitive was introduced by Desmedt and Frankel (CRYPTO 1989) in their context of “threshold cryptography.” The expansion factor is the total number of group elements in a full sharing divided by n. For threshold BBSS with t-privacy (\(1\le t \le n-1\)), \(t+1\)-reconstruction and arbitrary n, constructions with minimal expansion \(O(\log n)\) exist (CRYPTO 2002, 2005).

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

Given a finite group G, the solubilizer of an element x, denoted by Sol G ( x ) , is the set of all elements y such that 〈 x , y 〉 is a soluble subgroup of G. In this paper, we provide a classification for all solubilizers of elements in minimal simple groups. We also examine these sets to explore their properties by discussing some computational methods and making some conjectures for further work.

1. In studying the structure of abstract groups, one is often confronted with the seemingly trivial question: when do we consider a group (or class of groups) as known? The question has many aspects, and is usually a very complex one. Of course, the group is formally known as soon as its multiplication table is determined; but in most cases, little further information can be obtained from the multiplication table itself on the structural elements of the group, such as subgroups, normal subgroups, automorphisms, etc. If one is however, interested in the classification of groups belonging to a certain class, then the problem is concentrated on the enumeration of groups having some common property, and the structural elements of the individual groups are of a secondary importance. In this case, the problem can be formulated in the following way: Find some finite algebraical procedure, depending on some integral or more general algebraical parameters, which enables us to construct, in an abstract or symbolic way, each group belonging to the class under consideration. A group is of course constructed if its elements are known, and the groupproduct of any two elements can be calculated by means of the defining procedure. It is desirable that the parameters, called invariants of the group, should be characteristic of the group in the sense that no two distinct systems of operations should define the same group. A classical example for such an interpretation of the structure problem is furnished by the fundamental theorem of finite Abelian groups. In this paper we are concerned with the enumeration of certain finite metabelian groups. (Metabelian is a group with Abelian commutator subgroup.) The problem in all its generality is a very difficult one, and far from being solved in the above sense. However, special classes of metabelian groups are accessible to investigation, and such investigations have an interest in so far as they clear more aspects of the general problem. In studying the structure of metabelian groups, the method of group extensions is often found to be the most natural tool. As an extensive use will be made of this method in this paper, I want to formulate it in such a way as it seems most suitable for the case discussed here. A similar but formally different formulation was given by A. Scholz1. See also W. Magnus2. Let ( be a finite metabelian group, additively written, 2t an Abelian normal subgroup of (M such that the quotient group e = (5/Wt is Abelian. Elements

We consider various generalizations of the effective exponentiation problem, also known as the addition chains problem in additive terminology. We attempt to evaluate and consistently compare the results and the methods that were used to obtain them. The considered generalizations are complexity problems (i.e. finding the minimum number of required operations) for the computation a monomial of many variables, a set of powers, elements of finite Abelian groups, elements of free semigroups, systems of monomials, systems of integer linear forms, systems of elements of free Abelian groups along with some other problems. For all these directions of research we overview both the classical results and the results of the author and his followers.

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mapping each element of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the unique maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sortable element below it in the weak order. The fibers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> essentially define the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems for finite simple groups have been studied recently, and in this paper we study them for finite quasisimple groups G. We show that for a fixed non-trivial group word w and large enough G we have w(G)^3=G, namely every element of G is a product of 3 values of w. For various families of finite quasisimple groups, including covers of alternating groups, we obtain a stronger result, namely w(G)^2=G. However, in contrast with the case of simple groups, we show that w(G)^2 need not equal G for all large G. If k>2 then x^k y^k fails to be surjective for infinitely many finite quasisimple groups. The case k=2 turns out to be exceptional. Indeed, our last result shows that every element of a finite quasisimple group is a product of two squares. This can be regarded as a non-commutative analogue of Lagrange's four squares theorem.

Degenerations and orbits in finite abelian groups

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function fG(x1, . . . , xn) = x1 ċ x2 . . . xn in an arbitrary finite group (G, ċ), where the input of party Pi is xi ∈ G for i = 1, . . . , n. For flexibility, we are interested in protocols for fG which require only black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our results are as follows. First, on the negative side, we show that if (G, ċ) is non-abelian and n ≥ 4, then no ⌈n/2⌉-private protocol for computing fG exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for fG based on k-of-k threshold secret sharing schemes, which are efficiently implementable over any black-box group G. We reduce the problem of constructing such protocols to a combinatorial colouring problem in planar graphs. We then give two constructions for such graph colourings. Our first colouring construction gives a protocol with optimal collusion resistance t < n/2, but has exponential communication complexity O(n(2t+1/t)2) group elements (this construction easily extends to general adversary structures). Our second probabilistic colouring construction gives a protocol with (close to optimal) collusion resistance t < n/µ for a graph-related constant µ ≤ 2.948, and has efficient communication complexity O(nt2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

Magnetic materials are of great importance for energy harvesting and conversion and information technologies, as exemplified by permanent magnets and spintronic applications, respectively. There is a strong impetus to develop new high performance magnets to satisfy the increasing requirements of such technologies. On the other hand, high-throughput (HTP) screening based on density functional theory calculations provides an efficiency way to predict novel materials with target properties. Therefore, it is meaningful to accelerate the development of functional magnetic materials by using the high-throughput screening method. In this thesis, we endeavored to perform HTP engineering of both 3D and 2D magnetic materials, focusing on spintronic and permanent magnet applications. For the spintronic applications, based on HTP density functional theory calculations, we performed a systematic screening by examining peculiar electronic structure. For instance, we have done HTP screening for spin-gapless semiconductors (SGSs) in quaternary Heusler alloys XX'YZ. Following an empirical rule, we focused on compounds with 21, 26, or 28 valence electrons, resulting in 12000 possible chemical compositions. After systematically evaluating the thermodynamic, mechanical, and dynamical stabilities, we have identified 70 so far unreported SGSs, confirmed by explicit electronic structure calculations with proper magnetic ground states, of which 17 candidates have a distance to the convex hull smaller than 0.10 eV/atom. It is demonstrated that all four types of SGSs can be realized, defined based on the spin characters of the touching bands around the Fermi energy. Particularly, it is found that the type-II SGSs exhibit promising transport properties for spintronic applications, such as large anisotropic magnetoresistance and anomalous Nernst effects driven by spin-orbit coupling. On the other hand, as to 2D materials, we carried out a systematic HTP screening for in-plane ordered MXene (i-MXene), which can be obtained by etching the main group element (A) away from the recently synthesized in-plane ordered MAX (i-MAX) phase. Such 2D i-MXenes provide us a new playground for 2D magnetic materials. It is observed that the spin configurations, hence the magnetic anisotropy, can be tuned by strain. We found that five i-MXenes have a significantly large out-of-plane magnetic anisotropy energy (>0.5 meV/f.u.), and the four ferromagnetic candidates have high Curie temperature based on the 2D Ising model. Additionally, we found i-MXene can realize large Seebeck effect, antiferromagnetic topological insulator, and spin-gapless semiconductors, making them interesting for future studies. In terms of designing permanent magnets, we carried out a systematic HTP screening for rare-earth free permanent magnets by incorporating light interstitials (H, B, C, N) into magnetic full Heusler alloys. We successfully identified 32 candidates with an out-of-plane magnetic anisotropy larger than 0.4 MJ/m$^3$ as well as 10 cases with large in-plane anisotropy. Detailed analysis reveals that the interstitials are very efficient in inducing global tetragonal distortions, whereas the local chemical bonding and changes in the crystalline environment result in significant enhancement of MAE. We strongly believe this provides an efficient way to tailor MAE, and such newly predicted permanent magnet candidates are promising gap magnets between rare earth based Sm-Co and Nd-Fe-B and the transition metal based AlNiCo and ferrite. To summarise, in addition to screening over various types of functional magnetic materials, my work provides valuable solutions to the pending challenges of HTP design of magnetic materials. We have established automated workflows, which can be easily applied on the other magnetic materials. This paves the way for further design of advanced magnetic materials with optimal performance