Seminars in Finite Groups

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Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

Benjamin Steinberg: Contractibility of the orbit space of Brown’s p-subgroup complex – a new proof

5th May, 2023

K. Brown introduced in 1975 the p-subgroup complex of a finite group G. It is the simplicial complex whose vertices are the nontrivial p-subgroups of G, where a collection of subgroups spans a simplex if it is a chain. This complex was further studied by Quillen, who observed that for a finite group of Lie type G with defining characteristic p, this complex is homotopy equivalent to the building of G. He also conjectured that the p-subgroup complex is contractible if and only if G contains a nontrivial normal p-subgroup and proved his conjecture for solvable groups. The Quillen conjecture remains open but was proved for almost simple groups by Aschbacher and Kleidman, and strong reduction theorem was obtained by Aschbacher and Smith.

The group G acts on its p-subgroups by conjugation and hence acts simplicially on the p-subgroup complex. Webb conjectured in 1987 that the orbit space of the p-subgroup complex is always contractible. He proved that its mod-p homology vanishes using methods from group cohomology. Webb's conjecture was first proved by Symonds in 1998, and a number of other proofs have since appeared. All the proofs I am aware of go through Robinson's subcomplex, which is G-homotopy equivalent to Brown's. None of the proofs are explicit. Symonds computes the fundamental group and integral homology and uses the Hurewicz and Whitehead theorems. Bux gave an inductive  proof using a variant of Bestvina-Brady style discrete Morse theory. In this talk, I will use Brown's theory of collapsing schemes to give an explicit sequence of elementary collapses that collapses the orbit space of Robinson's subcomplex to the vertex corresponding to the conjugacy class of Sylow p-subgroups.

Pavel Turek: On stable modular plethysms of the natural module of SL2(𝔽p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Giada Volpato: On the restriction of a character of Sn to a Sylow p-subgroup

15th November, 2022

The relevance of the McKay Conjecture in the representation theory of finite groups has led to investigate how irreducible characters decompose when restricted to Sylow p-subgroups. In this talk we will focus on the symmetric groups. Since the linear constituents of the restriction to a Sylow p-subgroup has been studied a lot by E. Giannelli and S. Law, we will concentrate on constituents of higher degree. In particular, we will describe the set of the irreducible characters which allow a constituent of a fixed degree, separating the cases of p being odd and p=2.

Yifan Jing: Measure Growth in Compact Simple Lie Groups

8th November, 2022

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.

Pavel Shumyatsky: Commuting probability for subgroups of a finite group

4th November, 2022

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 indices [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.

Alexander Hulpke: Constructing Perfect Groups

2nd November, 2022

The construction of perfect groups of a given order can be considered as the prototype of the construction of insoluble groups of a given order. I will describe a recent project to enumerate, up to isomorphism, the perfect groups of order up to 2⋅106. It crucially relies on new tools for calculating cohomology, as well as improved implementations for isomorphism test. This work extends results of Holt and Plesken from 1989 and illustrates the scope of algorithmic improvements over the past decades.

Haralampos Geranios: On self-extensions of irreducible modules for symmetric groups

11th October, 2022

We work in the context of the modular representation theory of the symmetric groups. A long-standing conjecture, from the late 80s, suggests that there are no (non-trivial) self-extensions of irreducible modules over fields of odd characteristic. In this talk we will highlight several new positive results on this conjecture.

Nikolay Nikolov: On conjugacy classes of profinite groups

10th October, 2022

It is well-known that the number of conjugacy classes of a finite group G tends to infinity as the size of G tends to infinity. There is no such result for a general infinite group. In this talk I will discuss the situation when G is a profinite group and show that the number of conjugacy of G is then uncountable unless G is finite. The proof depends on many classical results on finite groups and in particular the classification of the finite simple groups.