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Robert Guralnick

Publications and source records attributed to Robert Guralnick.

At least 37 records · Page 2Linked to original sources
arXiv2015

Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representations.

math.GR↗
arXiv2015

Surjective word maps and Burnsides $p^aq^b$ theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map sending (x,y) to the product of the Nth powers of x and y is surjective on every finite non-abelian simple group; If is an odd integer, then the word map sending the triple (x,y,z) to the product of the Nth powers is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit-Thompson. We also prove asymptotic results about the surjectivity of the word map sending (x,y) to the product of the Nth powers of x and y that depend on the number of prime factors of the integer N.

math.GR↗
arXiv2015

Simple groups stabilizing polynomials

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f g = f$. In case $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we note that the subgroup of $GL(V)$ stabilizing $f$ often has identity component $G$ and we give applications realizing various groups, including the largest exceptional group $E_8$, as automorphism groups of polynomials and algebras. We show that starting with a simple group $G$ and an irreducible representation $V$, one can almost always find an $f$ whose stabilizer has identity component $G$ and that no such $f$ exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions $G < H \le SL(V)$ such that $V/H$ has the same dimension as $V/G$. The main results of this paper are new even in the special case where $k$ is the complex numbers.

math.GR↗
arXiv2015

Derangements in Subspace Actions of Finite Classical Groups

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

math.GR↗
arXiv2015

Adequate groups of low degree

The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G+-summands have dimension less than p (where G+ denotes the subgroup of G generated by all p-elements of G), are adequate.

math.RT↗
arXiv2014

On rational and concise words

A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G in X, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. It is still an open problem whether every word is concise in the class of residually finite groups. A word w is rational if the number of solutions to the equation w=g is the same as the number of solutions to w=g^e for every finite group G and for every e relatively prime to |G|. We observe that any rational word is concise in the class of residually finite groups. Further we give a sufficient condition for rationality of a word. This is used to produce a new family of words that are concise in the class of residually finite groups.

math.GR↗
arXiv2014

Expansion in finite simple groups of Lie type

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups.

math.GR↗
arXiv2013

Variations on the Baer--Suzuki Theorem

The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $\langle x, x^g \rangle$ is a $p$-group for all $g \in G$, then the normal closure of $x$ in $G$ is a $p$-group. We consider the case where $x^g$ is replaced by $y^g$ for some other $p$-element $y$. While the analog of Baer--Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of $p$-elements.

math.GR↗
arXiv2013

Products and commutators of classes in algebraic groups

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki theorem for finite groups. We also answer a question of Pavel Shumyatsky on commutators of pairs of conjugacy classes in simple algebraic groups. It turns out that the resulting examples are exactly those for which the product also consists of only finitely many classes.

math.GR↗
arXiv2012

Rational rigidity for E_8(p)

We prove the existence of certain rationally rigid triples E8 in good characteristic and thereby show that these groups over the prime field occur as Galois groups over the field of rational numbers. We show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0. As a byproduct we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a classification of possible overgroups in exceptional groups containing regular unipotent elements.

math.GR↗
arXiv2012

The Automorphism Groups of a Family of Maximal Curves

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n greater than or equal to 3, such that C_n is maximal over F_{q^{2n}}. In this paper, we determine the automorphism group Aut(C_n) when n > 3; in contrast with the case n=3, it fixes the point at infinity on C_n. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point. MSC:11G20, 14H37.

math.NT↗
arXiv2012

On the maximal number of coprime subdegrees in finite primitive permutation groups

The subdegrees of a transitive permutation group are the orbit lengths of a point stabilizer. For a finite primitive permutation group which is not cyclic of prime order, the largest subdegree shares a non-trivial common factor with each non-trivial subdegree. On the other hand it is possible for non-trivial subdegrees of primitive groups to be coprime, a famous example being the rank 5 action of the small Janko group on 266 points which has subdegrees of lengths 11 and 12. We prove that, for every finite primitive group, the maximal size of a set of pairwise coprime non-trivial subdegrees is at most 2.

math.GR↗
arXiv2011

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 acting completely reducibly on a vector space V: if the orbits containing the vectors a and b have coprime lengths m and n, we prove that the orbit containing a+b has length mn. Such groups H are always reducible if n and m are greater than 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. In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.

math.GR↗
arXiv2011

Lifting in Frattini Covers and A Characterization of Finite Solvable Groups

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1. This is also related to various questions about Fried's modular tower program and properties of Hurwitz spaces of covers of curves.

math.GR↗
arXiv2011

Adequate Subgroups II

The notion of adequate subgroups was introduced by Jack Thorne. It is a weakening of the notion of big subgroup used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to prove some new lifting theorems. In an appendix to Thorne's article, it was shown in that certain groups were adequate. One of the key aspects was the question of whether the span of the semsimple elements in the group is the full endomorphism ring of an absolutely irreducible module. We show that this is the case in prime characteristic p for p-solvable groups as long the dimension is not divisible by p. We also observe that the condition holds for certain infinite groups. Finally, we present the first examples showing that this condition need not hold and give a negative answer to a question of Richard Taylor.

math.RT↗