Finite Linear Groups Research Articles

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p 1, p 2, p 3 be primes. This is the second article in a series of three on the (p 1, p 2, p 3)-generation of the finite projective special unitary and linear groups PSU3(p n ), PSL3(p n ), where we say a noncyclic group is (p 1, p 2, p 3)-generated if it is a homomorphic image of the triangle group T p 1, p 2, p 3 . This paper is concerned with the case where p 1 = 2 and p 2 = p 3. We determine for any prime p 2 the prime powers p n such that PSU3(p n ) (respectively, PSL3(p n )) is a quotient of T = T 2, p 2, p 2 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU3(p n )) (respectively, Hom(T, PSL3(p n ))) is surjective as p n tends to infinity.

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p1, p2, p3 be primes. This is the final paper in a series of three on the (p1, p2, p3)-generation of the finite projective special unitary and linear groups PSU 3(pn), PSL 3(pn), where we say a noncyclic group is (p1, p2, p3)-generated if it is a homomorphic image of the triangle group Tp1, p2, p3 . This article is concerned with the case where p1 = 2 and p2 ≠ p3. We determine for any primes p2 ≠ p3 the prime powers pn such that PSU 3(pn) (respectively, PSL 3(pn)) is a quotient of T = T2, p2, p3 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU 3(pn)) (respectively, Hom(T, PSL 3(pn))) is surjective as pn tends to infinity.

Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory

Abstract Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proof of the integral K-theoretic Novikov conjecture, regarding split injectivity of the K-theoretic assembly map, for groups with finite decomposition complexity and finite CW models for their classifying spaces. By work of Guentner, Tessera, and Yu, this includes all (geometrically finite) linear groups.

Semigroups of polygons whose vertices define a centered partition of ℝ n

A partition \(\mathfrak{F}\) of a Euclidean space into finite subsets has subgroup property SP if the family of the convex hulls of the leaves of \(\mathfrak{F}\) constitutes a subgroup with respect to the Minkowski addition. If \(\mathfrak{F}\) consists of orbits of a finite linear groups then SP is equivalent to the fact that the group is a Coxeter group. In this article, this assertion is proved only under the assumption of continuity and centrality of \(\mathfrak{F}\) (this means that every leaf is inscribed in some sphere centered at zero). An example is given of a noncentered partition satisfying SP (such partitions cannot be Coxeter partitions).

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V) acting completely reducibly on a vector space V: if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n,m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor.

On Jordan decomposition of characters for [formula omitted

As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnaféʼs methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztigʼs twisted induction.

Almost recognizability by spectrum of finite simple linear groups of prime dimension

The spectrum of a group is the set of its element orders. Let L = PSLn(q), where n is a prime greater than 3. We show that every finite group whose spectrum is the same as the spectrum of L is isomorphic to an extension of L by a subgroup of the outer automorphism group of L.

Quotient-complete arc-transitive graphs

We examine the class of G-arc-transitive graphs with at least one nontrivial complete normal quotient, such that all other normal quotients are either complete or empty graphs. Such graphs arise naturally as basic graphs under normal quotient analysis on families of arc-transitive graphs. We construct all such graphs Γ, together with the corresponding group G≤Aut(Γ), which have 3 or more nontrivial complete G-normal quotients. In particular, we prove that these graphs have order the square of a prime power q, and arise from finite transitive linear groups H acting on a vector space of order q. The only disconnected graphs in this class are isomorphic to q copies of the complete graph on q vertices, and we classify all the connected graphs that arise from the transitive linear groups H where H is not a subgroup of a one-dimensional affine group ΓL(1,q). For the case where H≤ΓL(1,q) we construct some infinite families of examples. In most cases we determine which of the graphs constructed have diameter two.

A problem of Kollár and Larsen on finite linear groups and crepant resolutions

The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age ≤ 1 . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation. As a consequence of our main results, we derive some properties of symmetric spaces GU_{d}(\mathbb C)/G having shortest closed geodesics of bounded length, and of quotients \mathbb C^{d}/G having a crepant resolution.

On Invariants of Some Maximal Subgroups of Finite Classical Groups

The maximal subgroups of the finite classical groups are divided into nine classes by Aschbacher's theorem. In this paper, we give explicit transcendental bases of the invariant subfields of those maximal subgroups of classical groups of linear, symplectic and unitary cases that lie in the first two of these classes. Also, we show that the invariant rings of the maximal subgroups of the finite linear groups that lie in the first class are polynomial rings.

Finite-dimensional vertex algebra modules over fixed point differential subfields

Let K be a differential field over C with derivation D, G a finite linear automorphism group over K which preserves D, and KG the fixed point subfield of K under the action of G. We show that every finite-dimensional vertex algebra KG-module is contained in some twisted vertex algebra K-module.

Finite-dimensional vertex algebra modules over fixed point commutative subalgebras

Let A be a connected commutative C -algebra with derivation D, G a finite linear automorphism group of A which preserves D, and R = A G the fixed point subalgebra of A under the action of G. We show that if A is generated by a single element as an R-algebra and is a Galois extension over R in the sense of M. Auslander and O. Goldman, then every finite-dimensional indecomposable vertex algebra R-module has a structure of twisted vertex algebra A-module.

Bounding the size of permutation groups and complex linear groups of odd order

We provide improved bounds for the order of finite odd order permutation groups and complex linear groups, expressed in terms of the smallest prime divisor of the group order.

Primitivity testing of finite nilpotent linear groups

AbstractWe describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm inMagmais publicly available.

Congruence subgroups of the minimal covolume arithmetic Kleinian group

We consider the problem of finding the normal subgroups of the orientation preserving subgroup Δ+ of the [3,5,3]-Coxeter group with the factor group isomorphic to $\operatorname{\mathrm{PSL}}_{2}(\mathbb {F}_{q})$ . We identify all such groups with particular congruence subgroups of an arithmetic subgroup of PSL 2(ℂ) derived from a quaternion algebra over a quartic field. The result can be interpreted as a generalization of the Macbeath’s result on the classification of finite linear groups as Hurwitz groups to 3-dimensional hyperbolic space.

Conditions for solvability in complex p-solvable linear groups

We present conditions on the structure and degree n of a finite irreducible complex linear group that guarantee its solvability. In particular, we show that if such a group is p -solvable but not p -closed for some prime number p , then the group is solvable whenever n ≡ ± 1 ( mod p ) and n is also smaller than certain bounds which are on the order of p 2 .

Irreducibility testing of finite nilpotent linear groups

We describe an algorithm for irreducibility testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For a reducible group, our algorithm constructs a proper submodule. An implementation in Magma is publicly available.

Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums

In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via the Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi- dimensional Kloosterman sums in terms of the frequencies of weights in C(SL(n, q)). In particular, when n = 2, this gives a recursive formula for the power moments of Kloosterman sums. We illustrate our results with some examples.

Primitive parabolic permutation representations of finite special linear and unitary groups

The ranks, degrees, subdegrees, and double stabilizers of permutation representations of finite special linear and unitary groups on cosets of parabolic maximal subgroups are found.

On the orbit space of a compact linear Lie group with commutative connected component

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold. The topological quotient of a finite linear group was studied by Mikhailova in 1984. Here the connected component of the original Lie group G is assumed to be a torus of positive dimension.