Let G be a primitive permutation group on a finite set X and recall that a subset of X is a base for G if its pointwise stabiliser is trivial. The base size of G, denoted b(G), is defined to be the minimal size of a base. This natural invariant has been intensively studied for many years, finding a wide range of applications. In this talk I will report on recent progress concerning a project initiated by Jan Saxl in the 1990s, which seeks to determine all the primitive groups with b(G) = 2. I will also outline some of the main applications and I will highlight one or two related problems.
By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.
I will discuss joint work with Bob Guralnick and Russ Woodroofe. We investigate invariable generation of finite simple groups by two elements of prime or prime power order. We apply our results to a problem raised by Ken Brown on the topology of the order complex of the poset of all cosets of all proper subgroups of an arbitrary finite group.
In this talk, I will introduce the notion of functorial equivalence of blocks of finite groups, developed in recent joint work with Deniz Yilmaz. For a commutative ring R, and a field k of characteristic p>0, we introduce the category of diagonal p-permutation functors over R. To a pair (G,b) of a finite group G and a block idempotent b of kG, we associate a diagonal p-permutation functor FG,b, and we say that two such pairs (G,b) and (H,c) are functorially equivalent over R if the functors FG,b and FH,c are isomorphic. We show that the category of diagonal p-permutation functors over an algebraically closed field of characteristic 0 is semisimple. We obtain a precise description of the simple functors, and explicit formulas for their multiplicities as summands of FG,b. It follows that functorial equivalence preserves the defect groups of blocks, their number of simple modules, and their number of ordinary irreducible characters. This also leads to characterizations of nilpotent blocks, and to a finiteness theorem in the spirit of Donovan's finiteness conjecture.
The Picard group Tk(G) of the stable module category of a finite group has been an important object of study in modular representation theory, starting with work Dade in the 1970s. Its elements are equivalence classes of so-called endotrivial modules, i.e., modules M such that End(M) is isomorphic to a trivial module direct sum a projective kG-module. 1-dimensional characters, and their shifts, are examples of such modules, but often exotic elements exist as well. My talk will be a guided tour of how to calculate Tk(G), using methods from homotopy theory. The tour will visit joint work with Tobias Barthel and Joshua Hunt, with Jon Carlson, Nadia Mazza and Dan Nakano, and with Achim Krause.
If K is a subgroup of a finite group G, the probability that an element of G commutes with an element of K is denoted by Pr(K,G). The probability that two randomly chosen elements of G commute is denoted by Pr(G). A well known theorem, due to P. M. Neumann, says that if G is a finite group such that Pr(G) ≥ ε, then G has a nilpotent normal subgroup T of class at most 2 such that both the index [G:T] and the order |[T,T]| are ε-bounded.
In the talk we will discuss a stronger version of Neumann's theorem: if K is a subgroup of G such that Pr(K,G) ≥ ε, then there is a normal subgroup T ≤ G and a subgroup B ≤ K such that the indexes [G:T] and [K:B] and the order of the commutator subgroup [T,B] are ε-bounded.
We will also discuss a number of corollaries of this result. A typical application is that if in the above theorem K is the generalized Fitting subgroup F*(G), then G has a class-2-nilpotent normal subgroup R such that both the index [G:R] and the order of the commutator subgroup [R,R] are ε-bounded.
The McKay conjecture asserts that a finite group has the same number of odd degree irreducible characters as the normalizer of a Sylow 2-subgroup. The Alperin-McKay (A-M) conjecture generalizes this to the height-zero characters in 2-blocks.
In his original paper, McKay already showed that his conjecture holds for the finite symmetric groups Sn. In 2016, Giannelli, Tent and the speaker established a canonical bijection realising A-M for Sn; the height-zero irreducible characters in a 2-block are naturally parametrized by tuples of hooks whose lengths are certain powers of 2, and this parametrization is compatible with restriction to an appropriate 2-local subgroup.
Now corresponding to a 2-block of the symmetric group Sn, there is a 2-block of a maximal Young subgroup of Sn of the same weight. An obvious question is whether our canonical bijection is compatible with restriction of height-zero characters between these blocks.
Attempting to prove this compatibility lead me to formulate a conjecture asserting the Schur-positivity of certain differences of skew-Schur functions. The corresponding skew-shapes have triangular inner-shape, but otherwise do not refer to the 2-modular theory. I will describe my conjecture and give positive evidence in its favour.
In a recent paper, joint with Tobias Barthel and Drew Heard, we develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral G-Mackey functors for all finite groups G. In this talk, I will provide an introduction to the problem of classifying thick and localizing tensor-ideals via theories of support, describe in broad strokes some of the highlights of our theory (which builds on the work of Balmer-Favi, Stevenson, and Benson-Iyengar-Krause) and, time-permitting, discuss our applications in equivariant homotopy theory. The starting point for these equivariant applications is a recent computation (joint with Irakli Patchkoria and Christian Wimmer) of the Balmer spectrum of the category of derived Mackey functors. We similarly study the Balmer spectrum of the category of E(n)-local spectral Mackey functors, and harness our geometric theory of stratification to classify the localizing tensor-ideals of both categories.
We work in the group algebra kG of a finite group scheme defined over a field of characteristic p > 0. The stable category stmod(kG) of finitely generated kG-modules is tensor triangulated category of compact objects. Associated to any thick tensor ideal subcategory 𝒞 is a distinguished triangle → E → k → F → where E and F are idempotent modules in the stable category StMod(kG) of all kG-modules. Tensoring with F is the localization functor associated to 𝒞. Indeed, the endomorphism ring of F is the endomorphism ring of the trivial module in the localized category. In this lecture, we will discuss some results on the nature of the endomorphism ring of the module F and some strange variant support varieties for modules.
Let G be a finite simple group and S a normal subset of G. If |G| is large enough in terms of |S|/|G|, can we deduce that every element of G can be expressed as x y-1 for x and y elements of S? Shalev, Tiep, and I have proven that this is true assuming G is an alternating group or a group of Lie type in bounded rank, but the question remains open for classical groups of high rank over small fields. I will say something about the methods of proof, which involve both character methods and geometric ideas and also say something about the more general question of covering G by ST where S and T are large normal subsets.
In this talk we will discuss the structure of maximal subgroups of finite simple groups, particularly groups of Lie type. We will discuss subgroups of exceptional groups of Lie type, and a version of Ennola duality that exists for groups of Lie type, which relates untwisted and twisted groups of Lie type.
This is intended to complement the recent talk of Pham Huu Tiep in this seminar but will not assume familiarity with that talk. The estimates in the title are upper bounds of the form |χ(g)|≤χ(1)α, where χ is irreducible and α depends on the size of the centralizer of g. I will briefly discuss geometric applications of such bounds, explain how probability theory can be used to reduce to the case of elements g of small centralizer, discuss the level theory of characters, and conclude with the reduction to the case of characters χ of large degree. For such pairs (g,χ), exponential character bounds are trivial.
Zoltan Halasi: Babai’s conjecture for classical groups with generating sets containing transvections
A well-known conjecture of Babai states that if G is any finite simple group and X is a generating set of G, then the diameter of the Cayley graph Cay(G, X) is bounded by a polylogarithmic function of |G|. The goal of the talk is to sketch a proof of such a bound in the case that X contains a transvection.
Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We first discuss some such applications. Then we will report on recent results that produce such character bounds.
Babai's conjecture asserts that the diameter of the Cayley graph of any finite simple group G is bounded by (log |G|)O(1). This conjecture has been resolved for groups of bounded rank, but for groups of unbounded rank such as SLn(2) it is wide open. Even for random generators, only the case of alternating groups is resolved. In this talk we sketch the proof of Babai's conjecture for SLn(p), p = O(1), with at least three random generators. The proof extends to other classical groups over 𝔽q if we have at least q100 random generators. The heart of the proof consists of showing that the Schreier graph of SLn(q) acting on 𝔽qn with respect to q100 random generators is an expander graph.