Papers by Robert Guralnick
arXiv (Cornell University), Dec 30, 2015
We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched co... more We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all Oort groups lie in a particular class of finite groups that we characterize, with equality of classes under a conjecture about local liftings. We prove this equality unconditionally if the order of G is not divisible by 2p 2. We also treat the local lifting problem and relate it to the global problem.
European Journal of Mathematics
We prove some results about closures of certain matrix varieties consisting of elements with the ... more We prove some results about closures of certain matrix varieties consisting of elements with the same centralizer dimension. This generalizes a result of Dixmier and has applications to topological generation of simple algebraic groups.
Proceedings of the American Mathematical Society, 2002
The automorphism group of a free group Aut(F k) acts on the set of generating k-tuples (g 1 ,. ..... more The automorphism group of a free group Aut(F k) acts on the set of generating k-tuples (g 1 ,. .. , g k) of a group G. Higman showed that when k = 2, the union of conjugacy classes of the commutators [g 1 , g 2 ] and [g 2 , g 1 ] is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for k ≥ 3.
Annals of Mathematics, 2021
A group G is said to be 3 2-generated if every nontrivial element belongs to a generating pair. I... more A group G is said to be 3 2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ∈ G, there exists y ∈ G such that G = x1, y = x2, y. In other words, s(G) 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
Proceedings of the American Mathematical Society, 2006
We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of... more We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.
Israel Journal of Mathematics, 2020
Suppose that p is an odd prime and G is a finite group having no normal non-trivial p ′-subgroup.... more Suppose that p is an odd prime and G is a finite group having no normal non-trivial p ′-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. Contents 1. Introduction 1 2. Notation and Preliminary Results 3 3. The Z * p Theorem and a Proof 4 4. Almost Simple Groups 7 5. Second Proof of the Theorem and Proof of the Corollary 9 6. A permutation version of the Z * p theorem 10 References 11
Linear Algebra and its Applications, 2018
This note is concerned with isometries on the spaces of selfadjoint traceless matrices. We comput... more This note is concerned with isometries on the spaces of selfadjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten p-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.
Advances in Mathematics, 2016
We prove the existence of certain rationally rigid triples in F4(p) for good primes p (i.e., p > ... more We prove the existence of certain rationally rigid triples in F4(p) for good primes p (i.e., p > 3), thereby showing that these groups occur as regular Galois groups over Q(t) and so also over Q. 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.
Pacific Journal of Mathematics, 1997
Journal of Group Theory, 2017
Answering a question of Geoff Robinson, we compute the large n limiting proportion of i GL ( n ... more Answering a question of Geoff Robinson, we compute the large n limiting proportion of i GL ( n , q ) / q ⌊ n 2 / 2 ⌋ {i_{\mathrm{GL}}(n,q)/q^{\lfloor n^{2}/2\rfloor}} , where i GL ( n , q ) {i_{\mathrm{GL}}(n,q)} denotes the number of involutions in the group GL ( n , q ) {\mathrm{GL}(n,q)} . We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new “sum = product” identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.
Let Nn be the set of nilpotent n by n matrices over an algebraically closed field k. For each r ≥... more Let Nn be the set of nilpotent n by n matrices over an algebraically closed field k. For each r ≥ 2, let Cr(Nn) be the variety consisting of all pairwise commuting r-tuples of nilpotent matrices. It is well-kown that C2(Nn) is irreducible for every n. We study in this note the reducibility of Cr(Nn) for various values of n and r. In particular it will be shown that the reducibility of Cr(gl n), the variety of commuting r-tuples of n by n matrices, implies that of Cr(Nn) under certain condition. Then we prove that Cr(Nn) is reducible for all n, r ≥ 4. The ingredients of this result are also useful for getting a new lower bound of the dimensions of Cr(Nn) and Cr(gl n). Finally, we investigate values of n for which the variety C3(Nn) of nilpotent commuting triples is reducible.
The Electronic Journal of Linear Algebra, 2003
Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a... more Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q) numerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed inertia. The treatment of all these cases is unified, and is based on general group theoretic results concerning multiplicative maps of general and special linear groups, which in turn are based on classical results by Borel-Tits. Multiplicative maps that leave invariant elementary symmetric functions of eigenvalues and spectra are described also for matrices over a general commutative field.
Proceedings of the American Mathematical Society, 2015
Let G be a finite group and let K be the conjugacy class of x ∈ G. If K 2 is a conjugacy class of... more Let G be a finite group and let K be the conjugacy class of x ∈ G. If K 2 is a conjugacy class of G, then [x, G] is solvable. If the order of x is a power of prime, then [x, G] has a normal p-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties.
Journal of Algebra, 2015
A group-word w is called concise if whenever the set of w-values in a group G is finite it always... more 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 ∈ 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(x 1 ,. .. , x k) = g is the same as the number of solutions to w(x 1 ,. .. , x k) = 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. As a corollary we deduce that the word w = [.. . [x n1 1 , x 2 ] n2 ,. .. , x k ] n k is concise in the class of residually finite groups.
Journal of Algebra, 2014
Link to publication on Research at Birmingham portal Publisher Rights Statement: Articles publish... more Link to publication on Research at Birmingham portal Publisher Rights Statement: Articles published under an Elsevier user license are protected by copyright. Users may access, download, copy, translate, text and data mine (but may not redistribute, display or adapt) the articles for non-commercial purposes provided that users: Cite the article using an appropriate bibliographic citation (i.e. author(s), journal, article title, volume, issue, page numbers, DOI and the link to the definitive published version on ScienceDirect) Maintain the integrity of the article Retain copyright notices and links to these terms and conditions so it is clear to other users what can and cannot be done with the article Ensure that, for any content in the article that is identified as belonging to a third party, any re-use complies with the copyright policies of that third party General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. • Users may freely distribute the URL that is used to identify this publication. • Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. • User may use extracts from the document in line with the concept of 'fair dealing' under the Copyright, Designs and Patents Act 1988 (?) • Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.
Transactions of the American Mathematical Society, 1994
In this article, we attempt to generalize the result that for a commutative ring R R the outer au... more In this article, we attempt to generalize the result that for a commutative ring R R the outer automorphism group of R R -automorphisms of M n ( R ) {M_n}(R) is abelian of exponent n n . It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.
Proceedings of the American Mathematical Society, 1989
It is shown that if G G is a primitive permutation group on a set of size n n , then any abelian ... more It is shown that if G G is a primitive permutation group on a set of size n n , then any abelian quotient of G G has order at most n n . This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if M / K M/K is a minimal extension and L / K L/K is an abelian extension contained in the normal closure of M M , then the degree of L / K L/K is at most the degree of M / K M/K .
Pacific Journal of Mathematics, 1985
Linear Algebra and its Applications, 2004
We show that if k is an algebraically closed field and G a not necessarily connected reductive li... more We show that if k is an algebraically closed field and G a not necessarily connected reductive linear algebraic group over k, then G(k) is solvable, nilpotent or abelian if and only if every finite subgroup of G(k) is solvable, nilpotent or abelian respectively. We also obtain the analogous result for compact subgroups of GL n (C).
Journal of the European Mathematical Society, 2011
All finite simple groups of Lie type of rank n over a field of size q, with the possible exceptio... more All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G 2 (q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, A n and S n have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(log n + log q).
Uploads
Papers by Robert Guralnick