Classification Of Simple Groups Research Articles

On finite groups isospectral to simple classical groups

The spectrumω(G) of a finite group G is the set of element orders of G. Finite groups G and H are isospectral if their spectra coincide. Suppose that L is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic p. It is proved that a finite group G isospectral to L cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from p. Together with a series of previous results this implies that every finite group G isospectral to L is ‘close’ to L. Namely, if L is a linear or unitary group, then L⩽G⩽AutL, in particular, there are only finitely many such groups G for given L. If L is a symplectic or orthogonal group, then G has a unique nonabelian composition factor S and, for given L, there are at most 3 variants for S (including S≃L).

On control of the prime spectrum of a finite simple group

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

Metric ultraproducts of finite simple groups

Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter ω on N and a sequence (Gi)i∈N of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω-limit of the characteristics of the groups Gi. Moreover, G is a complete path-connected group with respect to the natural metric on G.

Block transitive 2−(v, 13,1) designs and classical simple groups

Block transitive 2−(v, 13,1) designs and classical simple groups

Base sizes for S-actions of finite classical groups

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.

Abundant p-singular elements in finite classical groups

In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of |G| distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of (1−1/p)/e, where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain ‘large’ subspaces of the natural G-module. We find explicit upper and lower bounds for the proportion of p-abundant elements in G, and prove that it approaches a (positive) limiting value as the dimension of G tends to infinity. It turns out that the limiting proportion of p-abundant elements is at least a constant multiple of the Isaacs–Kantor–Spaltenstein lower bound for the proportion of all p-singular elements.

Thompson’s conjecture for Lie type groups E 7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E 7(q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E 7(q)) is necessarily isomorphic to E 7(q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.

Counting conjugacy classes of elements of finite order in Lie groups

Using combinatorial techniques, we answer two questions about simple classical Lie groups. Define N(G,m) to be the number of conjugacy classes of elements of finite order m in a Lie group G, and N(G,m,s) to be the number of such classes whose elements have s distinct eigenvalues or conjugate pairs of eigenvalues. What is N(G,m) for G a unitary, orthogonal, or symplectic group? What is N(G,m,s) for these groups? For some cases, the first question was answered a few decades ago via group-theoretic techniques. It appears that the second question has not been asked before; here it is inspired by questions related to enumeration of vacua in string theory. Our combinatorial methods allow us to answer both questions.

On Weyl groups in minimal simple groups of finite Morley rank

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type.

Quasisimple classical groups and their complex group algebras

Let H be a finite quasisimple classical group, i.e., H is perfect and S:= H/Z(H) is a finite simple classical group. We prove that, excluding the open cases when S has a very exceptional Schur multiplier such as PSL3(4) or PSU4(3), H is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.

Nilpotent injectors in minimal simple groups

An odd nilpotent injector of a finite group G is defined to be a subgroup which is maximal subject to being nilpotent of odd order and containing a subgroup of maximal order amongst all abelian subgroups of odd order. We prove that the odd nilpotent injectors of a minimal simple group are all conjugate, extending the result from soluble groups. The proof does not use the classification of minimal simple groups.

Extremely primitive sporadic and alternating groups

A non-regular primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for An (or Sn); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1, … , n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co2 on the cosets of McL.

Simple classical groups of Lie type are determined by their character degrees

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G)={χ(1):χ∈Irr(G)} and let X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be a finite nonabelian simple classical group. In this paper, we will show that if G is a finite group and X1(G)=X1(H) then G is isomorphic to H. In particular, this implies that the nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras.

Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the finite simple group classification

Davenport’s Problem asks: What can we expect of two polynomials, over Z, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, Lewis and Schinzel. By bounding the degrees, but expanding the maps and variables in Davenport’s Problem, Galois stratification enhanced the separated variable theme, solving an Ax and Kochen problem from their Artin Conjecture work. Denef and Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement. By restricting the variables, but leaving the degrees unbounded, we found the striking distinction between Davenport’s problem over Q, solved by applying the Branch Cycle Lemma, and its generalization over any number field, solved by using the simple group classification. This encouraged Thompson to formulate the genus 0 problem on rational function monodromy groups. Guralnick and Thompson led its solution in stages. We look at two developments since the solution of Davenport’s problem. We use these to consider: Going beyond the simple group classification to handle imprimitive groups, and what is the role of covers and correspondences in going from algebraic equations to zeta functions with Chow motive coefficients.

On the problem of classifying simple compact quantum groups

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the author's first paper on simple quantum groups are shown to be simple using results of Raum and Weber.

Extremely primitive classical groups

A primitive permutation group is said to be extremely primitive if it is not regular and a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In this paper, we determine the examples in the case of almost simple classical groups. They comprise the 2-transitive actions of PSL2(q) and its extensions of degree q+1, and of Sp2m(2) of degrees 22m−1±2m−1, together with the 3/2-transitive actions of PSL2(q) on cosets of Dq+1, with q+1 a Fermat prime. In addition to these three families, there are four individual examples.

Computing nilpotent and unipotent canonical forms: a symmetric approach

AbstractLet k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

Bounding the order of a group with a large character degree

Let d be the degree of some irreducible character of a finite group G. We can write | G | = d ( d + e ) for some nonnegative integer e, and we show that if e > 1 , then | G | ⩽ B e 6 for some universal constant B. This result, which improves a non-polynomial bound of Snyder, relies on recent work of Larsen, Malle and Tiep on character degrees of simple groups, and so it uses the simple group classification.

On Regular Orbits of Elements of Classical Groups in Their Permutation Representations

Let H be a simple finite classical group not isomorphic to PSL(n, q) for any n, q. We prove that every cyclic subgroup of H has a regular orbit on every nontrivial permutation H-set.

On spectra of covers of finite simple classical groups

On spectra of covers of finite simple classical groups