Adequate subgroups
We study adequate subgroups of $GL_n$ over a finite field. This notion is useful in the study of automorphy lifting theorems. In particular, we give a sufficient condition for a subgroup to be adequate.
math.NT↗
arXiv subjects
Robert Guralnick
Publications and source records attributed to Robert Guralnick.
A new solvability criterion for finite groups
In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.
math.GR↗
Strongly dense free subgroups of semisimple algebraic groups
We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type.
math.GR↗
Products of conjugacy classes and fixed point spaces
We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).
math.GR↗
Simple groups admit Beauville structures
We answer a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type and obtain definitive results about the variety of triples in semisimple regular classes with product 1. Finally, we prove that any finite simple group contains two conjugacy classes C,D such that any pair of elements in C x D generates the group.
math.GR↗
Representation Growth in positive characteristic and conjugacy classes of maximal subgroups
We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n is bounded by a low degree polynomial in n. As a consequence, we show that the number of conjugacy classes of maximal subgroups of a finite almost simple group G is at most O(log|G|).
math.RT↗
A problem of Kollar 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. Kollar and M. Larsen on the structure of finite irreducible linear groups generated by elements of age at most 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 for the unitayr group having shortest closed geodesics of bounded length, and of quotients of affine space by a finite group having a crepant resolution.
math.GR↗
On a subfactor generalization of Wall's conjecture
In this paper we discuss a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. We explore special cases of this conjecture and present supporting evidence. In particular we prove special cases of this conjecture related to some finite dimensional Kac Algebras of Izumi-Kosaki type which include relative version of Wall's conjecture for solvable groups.
math.OA↗
Certain subgroups of Weyl groups are split
Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.
math.RT↗
Low Degree Representations of Simple Lie Groups
We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.
math.RT↗
The local lifting problem for actions of finite groups on curves
Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $ϕ$ of a finite group $G$ on $k[[t]]$. To each such $ϕ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $ϕ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $ϕ$ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $ϕ$ due to Bertin vanishes for some $ϕ$, or for all sufficiently ramified $ϕ$. These results provide evidence for a strengthening of Oort's lifting conjecture.
math.AG↗
Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with Applications to Derangements
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).
math.GR↗
Characterizations of the Solvable Radical
We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs that do not use the Classification theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.
math.GR↗
Presentations of finite simple groups: a quantitative approach
Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $C$ such that $\dim H^2(G,M)\leq C\dim M$ for every finite simple group $G$, every prime $p$ and every irreducible $F_p [G]$-module $M$.
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Presentations of Finite Simple Groups: Profinite and Cohomological Approaches
We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.
math.GR↗
Oort groups and lifting problems
Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture.
math.AG↗
On fixed points of permutations
The number of fixed points of a random permutation of 1,2,...,n has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial -- almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of 1,2,...,n, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results.
math.CO↗
Frobenius-Schur Indicators for Subgroups and the Drinfel'd Double of Weyl Groups
If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgroup of the Weyl group W, then all representations of the centralizer of E in W are defined over the rational field.
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