Classification Of Finite Groups Research Articles

Diameter of classical groups generated by transvections

Let G be a finite classical group generated by transvections, i.e., one of SLn(q), SUn(q), Sp2n(q), or O2n±(q)(qeven), and let X be a generating set for G containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph Cay(G,X) is bounded by (nlog⁡q)C for some constant C. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection.By combining this with a result of the author and Jezernik it follows that if G is one of SLn(q), SUn(q), Sp2n(q) and X contains three random generators then with high probability the diameter Cay(G,X) is bounded by nO(log⁡q). This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.

О прямых произведениях групп диэдра в локально конечных группах

When studying infinite groups, as a rule, some finiteness conditions are imposed. For example, they require that the group be periodic, a Shunkov group, a Frobenius group, or a locally finite group. The concept of saturation allows us to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated by direct products of various groups. Significant progress has been made in solving the problem of B. Amberg and L. S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. It is proved that a locally finite group saturated by the direct product of a finite number of finite groups of a dihedron is isomorphic to the direct product of locally cyclic groups multiplied by an involution. It is also proved that a locally finite group saturated by the direct product of a finite number of finite dihedral groups is solvable.

Group action and its application on classification of low order groups

The classification of finite groups is an important topic in mathematics throughout history of mathematics. The topic of this paper is to use group action as a tool, to classify some special finite groups and some low order groups. First this paper introduces some concepts of group action. Then this paper states and proves some important theorems related to group action. For example, the Sylows theorem, which is very important in this paper. Research has found that, groups of specific order, such as groups whose order are 2p,p^2, pq(p, q are distinct prime numbers), p3(p is prime) can be classified using group action and the technique of semi-direct product, and groups whose order are no more than 15 are classified which can be seen as the special situations of the above ones. But in general, to make classification of a larger range of finite groups, more tools should be introduced.

Character levels and character bounds for finite classical groups

The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to |G|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$|G|$\\end{document} in a logarithmic sense.

Subgroups of classical groups that are transitive on subspaces

For each finite classical group G, we classify the subgroups of G which act transitively on a G-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick classical generalised quadrangles.

On properties of the lattice of all τ-closed n-multiply σ-local formations

Let be some partition of the set of all primes , G a finite group, a class of finite groups, and A function f of the form is called a formation σ-function. For any formation σ-function f the class is defined as follows: If for some formation σ-function f we have then the class is called σ-local and f is called a σ-local definition of Every formation is called 0-multiply σ-local. For a formation is called n-multiply σ-local provided either is the class of all identity groups or where is -multiply σ-local for all Let be a set of subgroups of G such that . Then τ is called a subgroup functor if for every epimorphism : and any groups and we have and . A class of groups is called τ-closed if for all . In this paper we prove that the lattice of all τ-closed n-multiply σ-local formations is an algebraic modular lattice of formations.

A constructive approach: From local subgroups to new classes of finite groups

Let G be a finite group and S be a proper subgroup of G. A group G is called an S-(S-quasinormal)-group if every local subgroup of G is either an S-quasinormal subgroup or conjugate to a subgroup of S. The main purpose of this construction is to demonstrate a new way of analyzing the structure of a finite group by the properties and the number of conjugacy classes of its local subgroups.

On the incomparability of systems of sets of lengths

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. We consider the system L(H) of all sets of lengths of H and study when L(H) contains or is contained in a system L(H′) of a Krull monoid H′ with finite class group G′, prime divisors in all classes and Davenport constant D(G′)=D(G). Among others, we show that if G is either cyclic of order m≥7 or an elementary 2-group of rank m−1≥6, and G′ is any group which is non-isomorphic to G but with Davenport constant D(G′)=D(G), then the systems L(H) and L(H′) are incomparable.

Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of gin textrm{SL}_n(q) is the product of le k pairwise distinct monic irreducible polynomials over mathbb {F}_q, then every element x of textrm{SL}_n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n=p is prime, if g has order frac{q^p-1}{q-1}, then every non-scalar element x in textrm{SL}_p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

On matrix units of semisimple group algebras

We present two methods to explicitly compute complete sets of orthogonal primitive idempotents in simple components of semisimple group algebras over finite Galois extensions of the rationals and some classes of finite groups, including finite nilpotent groups.

Локально конечные группы, насыщенные прямым произведением двух конечных групп диэдра

In the study of infinite groups, as a rule, some finiteness conditions are imposed. For example, the group is required to be periodic, Shunkov group, Frobenius group, locally finite group. The concept of saturation makes it possible to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated with direct products of various groups. In this paper, a partial solution to the problem of B. Amberg and L.S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. The structure of a locally finite group saturated with a direct product of two finite dihedral groups is established and it is proved that in this case the group is solvable. The result obtained is an important step towards solving the problem of Amberg and Kazarin.

A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

The spectrum ω(G) of a finite group G is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group G, there are four different primes such that ω(G) contains all their pairwise products but not a product of any three of these numbers, then G is nonsolvable. Using this result, we show that for q ⩾ 8 and q ≠ 32, the direct square Sz(q) × Sz(q) of the simple exceptional Suzuki group Sz(q) is uniquely characterized by its spectrum in the class of finite groups, while for Sz(32) × Sz(32), there are exactly four finite groups with the same spectrum.

On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group

Let \(G\) be a finite group. Its spectrum \(\omega(G)\) is the set of all element orders of \(G\). The prime spectrum \(\pi(G)\) is the set of all prime divisors of the order of \(G\). The Gruenberg–Kegel graph (or the prime graph) \(\Gamma(G)\) is the simple graph with vertex set \(\pi(G)\) in which any two vertices \(p\) and \(q\) are adjacent if and only if \(pq\in\omega(G)\). The structural Gruenberg–Kegel theorem implies that the class of finite groups with disconnected Gruenberg–Kegel graphs widely generalizes the class of finite Frobenius groups, whose role in finite group theory is absolutely exceptional. The question of coincidence of Gruenberg–Kegel graphs of a finite Frobenius group and of an almost simple group naturally arises. The answer to the question is known in the cases when the Frobenius group is solvable and when the almost simple group coincides with its socle. In this short note we answer the question in the case when the Frobenius group is nonsolvable and the socle of the almost simple group is isomorphic to \(PSL_{2}(q)\) for some \(q\).

Cutoff on Ramanujan complexes and classical groups

The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper we establish total-variation cutoff for simple random walk on Ramanujan complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, we obtain explicit generators for the finite classical groups $\mathrm{PGL}_{n}(\mathbb{F}_{q})$ for which the associated Cayley graphs exhibit total-variation cutoff.

On classification of finite groups with four generators three of which having orders $p,p,q$ $(p$ < $q)$. I.

On classification of finite groups with four generators three of which having orders $p,p,q$ $(p$ < $q)$. I.

On a certain class of finite groups with two independent generators

On a certain class of finite groups with two independent generators

On classification of finite groups with four generators three of which having prime orders $p,q,q$ $(p$ < $q)$. II.

On classification of finite groups with four generators three of which having prime orders $p,q,q$ $(p$ < $q)$. II.

Carbon-Based Metal Free Catalysts: Preparation, Structural and Morphological Property and Application

Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subspace of V, such that U∩U′=0 and U⊕U′ is non-degenerate (the sum U⊕U′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim⁡(U) and dim⁡(U′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U′) of non-degenerate subspaces and certain pairs (g,g′)∈G2 of group elements where U=im(g−1) and U′=im(g′−1).

Random generation of direct sums of finite non-degenerate subspaces

Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subspace of V, such that U∩U′=0 and U⊕U′ is non-degenerate (the sum U⊕U′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim⁡(U) and dim⁡(U′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U′) of non-degenerate subspaces and certain pairs (g,g′)∈G2 of group elements where U=im(g−1) and U′=im(g′−1).

Comparing the order and the minimal number of generators of a finite irreducible linear group

We denote the smallest size of a generating set of a group G by d(G). We prove that d(G)log⁡|G|=O(n2log⁡q) for irreducible subgroups G of GL(n,q), and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.