arXiv subjects

Robert M. Guralnick

Publications and source records attributed to Robert M. Guralnick.

At least 73 records · Page 4Linked to original sources

arXiv2016

Spinors and essential dimension

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2.

math.GR↗

arXiv2016

Finite Groups with Odd Sylow Normalizers

We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups.

math.GR↗

arXiv2015

Effective Results on the Waring Problem for Finite Simple Groups

Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x,y in G, x of prime order, and |y| is divisible by at most two primes, such that the product of the conjugacy classes of x and y contain all non-central elements of G. In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of one of the main results of Larsen, Shalev and Tiep by showing that, given any positive integer m, if the order of a finite simple group S is at least f(m) for a specified function f, then every element in S is a product of two mth powers. Furthermore, the verbal width of the mth power word on any finite simple group S is at most g(m) for a specified function g. We also show that, given any two non-trivial words v, w, if G is a finite quasisimple group of large enough order, then v(G)w(G) contains all non-central elements of G.

math.GR↗

arXiv2013

Reducibility of nilpotent commuting varieties

Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\N_n)$ under certain condition. Then we prove that $C_r(\N_n)$ is reducible for all $n, r\ge 4$. The ingredients of this result are also useful for getting a new lower bound of the dimensions of $C_r(\N_n)$ and $C_r(\mathfrak{gl}_n)$. Finally, we investigate values of $n$ for which the variety $C_3(\N_n)$ of nilpotent commuting triples is reducible.

math.RT↗

arXiv2013

On the non-coprime k(GV) problem

Let V be a finite faithful completely reducible FG-module for a finite field F and a finite group G. In various cases explicit linear bounds in |V| are given for the numbers of conjugacy classes k(GV) and k(G) of the semidirect product GV and of the group G respectively. These results concern the so-called non-coprime k(GV)-problem.

math.GR↗

arXiv2013

Bass units as free factors in integral group rings of simple groups

Let $G$ be a finite group, $u$ a Bass unit based on an element $a$ of $G$ of prime order, and assume that $u$ has infinite order modulo the center of the units of the integral group ring $\Z G$. It was recently proved that if $G$ is solvable then there is a Bass unit or a bicyclic unit $v$ and a positive integer $n$ such that the group generated by $u^n$ and $v^n$ is a non-abelian free group. It has been conjectured that this holds for arbitrary groups $G$. To prove this conjecture it is enough to do it under the assumption that $G$ is simple and $a$ is a dihedral $p$-critical element in $G$. We first classify the simple groups with a dihedral $p$-critical element. They are all of the form $\PSL(2,q)$. We prove the conjecture for $p=5$; for $p>5$ and $q$ even; and for $p>5$ and $q+1=2p$. We also provide a sufficient condition for the conjecture to hold for $p>5$ and $q$ odd. With the help of computers we have verified the sufficient condition for all $q<10000$.

math.GR↗

arXiv2012

Appendix A: Adequacy of representations of finite groups of Lie type

Thorne introduced the notion of adequate representations as a weakening of the big representations used by Wiles and Taylor and others. In this appendix to Dieulefait's paper, Automorphy of Symm5(GL(2)) and base change, we show that certain representations of SL(2,q) are adequate. This is used by Dieulefait to prove results about Hecke eigenforms of level 1 and newforms. We also prove some general results about adequacy for representations of finite groups of Lie type in the natural characteristic.

math.GR↗

arXiv2012

Cosets of Sylow p-subgroups and a Question of Richard Taylor

We prove that for any prime p there exist infinitely many finite simple groups G with a coset xP of a Sylow p-subgroup P of G such that every element of xP has order divisible by p. John Thompson proved this for p=2 in 1967 answering a question of Lowell Paige. This result is used to answer a question of Richard Taylor on adequate representations.

math.GR↗

arXiv2012

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5.

math.GR↗

arXiv2010

Frobenius subgroups of free profinite products

We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.

math.GR↗

arXiv2010

Average dimension of fixed point spaces with applications

Let $G$ be a finite group, $F$ a field, and $V$ a finite dimensional $FG$-module such that $G$ has no trivial composition factor on $V$. Then the arithmetic average dimension of the fixed point spaces of elements of $G$ on $V$ is at most $(1/p) \dim V$ where $p$ is the smallest prime divisor of the order of $G$. This answers and generalizes a 1966 conjecture of Neumann which also appeared in a paper of Neumann and Vaughan-Lee and also as a problem in The Kourovka Notebook posted by Vaughan-Lee. Our result also generalizes a recent theorem of Isaacs, Keller, Meierfrankenfeld, and Moretó. Various applications are given. For example, another conjecture of Neumann and Vaughan-Lee is proven and some results of Segal and Shalev are improved and/or generalized concerning BFC groups.

math.GR↗

arXiv2008

Symmetric powers and a problem of Kollar and Larsen

We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar.

math.RT↗

arXiv2008

Nonisomorphic curves that become isomorphic over extensions of coprime degrees

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field and let r and s be integers greater than 1 that are coprime to one another. We show that one can find a finite extension K of K_0, a degree-r extension L of K, a degree-s extension M of K, and two curves C and D over K such that C and D become isomorphic to one another over L and over M, but not over any proper subextensions of L/K or M/K. We show that such C and D can never have genus 0, and that if K is finite, C and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree extension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2 examples occur in every characteristic other than 3. Our detailed analysis of the case {r,s} = {2,3} shows that over every finite field K there exist nonisomorphic curves C and D that become isomorphic to one another over the quadratic and cubic extensions of K. Most of our proofs rely on Galois cohomology. Without using Galois cohomology, we show that two nonisomorphic genus-0 curves over an arbitrary field remain nonisomorphic over every odd-degree extension of the base field.

math.AG↗

arXiv2008

Polynomials with PSL(2) monodromy

Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.

math.AG↗

arXiv2008

A new family of exceptional polynomials in characteristic two

We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family.

math.NT↗

arXiv2008

Characteristic polynomials of automorphisms of hyperelliptic curves

Let alpha be an automorphism of a hyperelliptic curve C of genus g, and let alpha' be the automorphism of P^1 induced by alpha. Let n be the order of alpha and let n' be the order of alpha'. We show that the triple (g,n,n') completely determines the characteristic polynomial of the automorphism alpha^* of the Jacobian of C, unless n is even, n=n', and (2g+2)/n is even, in which case there are two possibilities. We give explicit formulas for the characteristic polynomial in all cases.

math.AG↗

arXiv2007

Exceptional covers and bijections on rational points

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively onto Y(F_q) if and only if f maps X(F_q) injectively into Y(F_q). Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory.

math.NT↗

arXiv2005

Orders of Finite Groups of Matrices

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields. For the most part, we work over an arbitrary number field rather than the rationals. The first half of the article is expository and is intended to be accessible to graduate students and advanced undergraduates. It gives a self-contained treatment, following Schur, over the field of rational numbers.

math.GR↗