A Class of Quasi-classical Groups over Finite Fields

We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non-abelian matrix group remains open. For the purposes of polynomial time computation, we reduce this problem to the following “affine case:” Let A be an elementary abelian p-group which is a non-central minimal normal subgroup of a finite group G. Assume further that the quotient G/A is a finite simple group S of Lie type of characteristic p. The algorithmic goal is to distinguish the simple group S from the nonsimple group G. Both S and G are given as “black-box groups of characteristic p.” The reduction to this basic problem involves a large number of recent techniques and results which we review along the way. We address the affine case for the groups S = PSL(2, q) where q = p , p prime. We show that PSL(2, p) can be recognized in Monte Carlo polynomial time among all black-box groups of known characteristic. The situation for f ≥ 2 seems much harder; we exhibit challenging open cases for every f ≥ 2 when q is large. Along with S = PSL(2, p), the positive result holds for all simple groups S of Lie type of characteristic p with the property that in every nontrivial S-module in characteristic p, every element of S has a fixed point. We call these groups “unisingular.” Very recently, Guralnick, Saxl, and Tiep classified these groups. They found that a large number of classes of simple groups are unisingular, including certain classes of linear and unitary groups, all symplectic groups over an odd prime field, all orthogonal groups of odd degree over an odd prime field, many orthogonal groups of even degree, and a number of classes of exceptional groups. 2 Laszlo Babai and Aner Shalev

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding finite-dimensional groups, as the rank tends to infinity. We solve a q-deformed version of the latter problem for orthogonal and symplectic groups, extending previously known results for the unitary group. The proof is based on novel determinantal and double-contour integral formulas for the q-specialized characters.

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

A supercharacter theory for involutive algebra groups

By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.

Let A be a linear transformation with minimum polynomial mA(t) and characteristic polynomial cA(t). We say that A is cyclic if mA(t) = cA(t), semisimple if mA(t) has no repeated factors, and separable if it is both cyclic and semisimple. The proportions of elements of GL(d, q) which are cyclic or separable have been investigated by, amongst others, Neumann and Praeger [7], Wall [10], and Fulman [4] who also studies semisimple elements. Making use of their results, we gave estimates in [1] for these proportions in SL(d, q). Fulman, Neumann and Praeger [5] have recently found good estimates for the corresponding proportions in the finite classical groups, U(d, q2), Sp(2m, q) and O±(d, q). They raise the question of whether similar methods may be used to calculate the proportions in various subgroups and supergroups of the classical groups. In a series of three papers, we shall show how this may be done. In this, the first paper, we deal with the special unitary groups SU(d, q2), consisting of unitary transformations with determinant 1. In the second paper [2] we shall focus on the conformal groups GSp(2m, q) and GO±(d, q), and in the third paper [3] we shall consider how the method may be extended to a number of groups of orthogonal type, including the special orthogonal group and the derived group of the orthogonal group. Our method in [1] involved the vector space cycle index of Kung [6] and Stong [8], which we describe briefly in Section 2. Similarly, in the current series

Let Γ be chosen from the orthogonal group O n according to Haar measure, and let A be an n x n real matrix with non-random entries satisfying TrAA* = n.We show that TrAΓ converges in distribution to a standard normal random variable as n -> oo uniformly in A. This extends a theorem of E. Borel.The result is applied to show that if entries βi, ,βk n are selected from Γ where k n -• oo as n -• oo, then J T~ Ί2\ k =i^ A, 0 < ί < 1 converges to Brownian motion.Partial results in this direction are obtained for the unitary and symplectic groups.

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

The list of known finite simple groups other than the cyclic, alternating, and Mathieu groups consists of the classical groups which are (projective) unimodular, orthogonal, symplectic, and unitary groups, the exceptional groups which are the direct analogues of the exceptional Lie groups, and certain twisted types which are constructed with the aid of Lie theory (see §§3 and 4 below). In this article, it is proved that each of these groups is generated by two of its elements. It is possible that one of the generators can be chosen of order 2, as is the case for the projective unimodular group (1), or even that one of the generators can be chosen as an arbitrary element other than the identity, as is the case for the alternating groups. Either of these results, if true, would quite likely require methods much more detailed than those used here.

if i < n and if k is any field with more than two elements. As discussed at this conference, no such stability theorem for other classical groups is known. In this paper, we give a prime-toresidue-characteristic homological stability theorem for classical groups over finite fields (Theorem 2), a stability theorem for the integral homology of classical groups over the algebraic closure of a finite field (Theorem 3), and a stability theorem for the integral homology of the special linear groups for finite fields with more than two elements (Theorem 6). We begin with the following well known stability results for the classical Lie groups. The proof is readily obtained by using , S 4n+3 ~ BSPn and the fibrations S 2n+l * BU n BUn+ 1 ~ BSPn+l, S n . BO n * BOn+ 1.

In Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434), a multiplicity one theorem is proved for general linear groups, orthogonal groups, and unitary groups ( $GL, O,$ and U) over p-adic local fields. That is to say that when we have a pair of such groups $G_n{\subseteq } G_{n+1}$ , any restriction of an irreducible smooth representation of $G_{n+1}$ to $G_n$ is multiplicity-free. This property is already known for $GL$ over a local field of positive characteristic, and in this paper, we also give a proof for $O,U$ , and $SO$ over local fields of positive odd characteristic. These theorems are shown in Gan, Gross, and Prasad (2012, Sur les Conjectures de Gross et Prasad. I, Société Mathématique de France) to imply the uniqueness of Bessel models, and in Chen and Sun (2015, International Mathematics Research Notice 2015, 5849–5873) to imply the uniqueness of Rankin–Selberg models. We also prove simultaneously the uniqueness of Fourier–Jacobi models, following the outlines of the proof in Sun (2012, American Journal of Mathematics 134, 1655–1678).By the Gelfand–Kazhdan criterion, the multiplicity one property for a pair $H\leq G$ follows from the statement that any distribution on G invariant to conjugations by H is also invariant to some anti-involution of G preserving H. This statement for $GL, O$ , and U over p-adic local fields is proved in Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434). An adaptation of the proof for $GL$ that works over of local fields of positive odd characteristic is given in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396). In this paper, we give similar adaptations of the proofs of the theorems on orthogonal and unitary groups, as well as similar theorems for special orthogonal groups and for symplectic groups. Our methods are a synergy of the methods used over characteristic 0 (Aizenbud et al. [2010, Annals of Mathematics 172, 1407–1434]; Sun [2012, American Journal of Mathematics 134, 1655–1678]; and Waldspurger [2012, Astérisque 346, 313–318]) and of those used in Mezer (2020, Mathematische Zeitschrift 297, 1383–1396).

Let \( \mathbb{F}_q \) be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over \( \mathbb{F}_q \) and \( \mathbb{F}_q \) be the rational function field over \( \mathbb{F}_q \). We seek to understand the structure of the rational invariant subfield \( \mathbb{F}_q \). In this paper, we prove that \( \mathbb{F}_q \) is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

The spectrum of a finite group G is the set of the orders of elements of G. In the present article, we obtain an arithmetical description of the spectra of finite symplectic and orthogonal groups. In particular, we describe the spectra of finite simple groups in these two classes.

We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.

The aim of the workshop was to give a panoramic view of the main lines of research on various aspects of growth and expansion in finite groups. The main topics included: diameter bounds for Cayley graphs of alternating and classical groups, with results for generating sets that can be arbitrary, random, or containing special elements; constructions of expander families of finite simple groups, with the use of property (T); and character bounds for finite simple groups, yielding applications to word map problems and random walks on Cayley graphs. A series of smaller topics explored connections and similarities to different problems in group theory.