Let q be a nontrivial odd prime power, and let $n \ge 2$ be a natural number with $(n,q) \ne (2,3)$. We characterize the groups $PSL_n(q)$ and $PSU_n(q)$ by their $2$-fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.

In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.

Let K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.

The Alperin–McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin–McKay conjecture holds if the so-called inductive Alperin–McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper, we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this, we obtain a proof of the Alperin–McKay conjecture for $2$-blocks of finite groups with abelian defect.

Let $\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$ be the character table for $S_n,$ where the indices $\lambda $ and $\mu $ run over the $p(n)$ many integer partitions of $n.$ In this note, we study $Z_{\ell }(n),$ the number of zero entries $\chi _{\lambda }(\mu )$ in $\mathcal {C}_n,$ where $\lambda $ is an $\ell $-core partition of $n.$ For every prime $\ell \geq 5,$ we prove an asymptotic formula of the form

$$ \begin{align*}Z_{\ell}(n)\sim \alpha_{\ell}\cdot \sigma_{\ell}(n+\delta_{\ell})p(n)\gg_{\ell} n^{\frac{\ell-5}{2}}e^{\pi\sqrt{2n/3}}, \end{align*} $$

$\sigma _{\ell }(n)$

$\delta _{\ell }:=(\ell ^2-1)/24,$

$1/\alpha _{\ell }>0$

$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$

$\ell $

$n>\ell ^6/24,$

$\chi _{\lambda }(\mu )=0$

$\lambda $

$\mu $

$\ell $

$Z^*_{\ell }(n)$

$\ell $

$\ell \geq 5$

$$ \begin{align*}Z^*_{\ell}(n)\sim \alpha_{\ell}^2 \cdot \sigma_{\ell}( n+\delta_{\ell})^2 \gg_{\ell} n^{\ell-3}. \end{align*} $$

Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $G=\mathrm {SO}(2n+1)$, $\mathrm {Sp}(2n)$, and $\mathrm {O}(2n)$.

The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we prove that, for any finite abelian group A, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of A.

We calculate the extension groups between simple modules of pro-p-Iwahori Hecke algebras.

We present a family of counterexamples to a question proposed recently by Moretó concerning the character codegrees and the element orders of a finite solvable group.

Let $\eta (G)$ be the number of conjugacy classes of maximal cyclic subgroups of G. We prove that if G is a p-group of order $p^n$ and nilpotence class l, then $\eta (G)$ is bounded below by a linear function in $n/l$.

Let $G$ be a finite group. An element $g \in G$ is called a vanishing element in $G$ if there exists an irreducible character $\chi$ of $G$ such that $\chi (g)=0$. The size of a conjugacy class of $G$ containing a vanishing element is called a vanishing conjugacy class size of $G$. In this paper, we give an affirmative answer to the problem raised by Bianchi, Camina, Lewis and Pacifici about the solvability of finite groups with exactly one vanishing conjugacy class size.

Let $\pi $ be a set of primes. We say that a group G satisfies $D_{\pi }$ if G possesses a Hall $\pi $-subgroup H and every $\pi $-subgroup of G is contained in a conjugate of H. We give a new $D_{\pi }$-criterion following Wielandt’s idea, which is a generalisation of Wielandt’s and Rusakov’s results.

No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than $4$, there are infinitely many groups with exactly k nonpower subgroups.

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$. In particular, we improve bounds found by Newman for the order of $M(2,2^k)$. We study identities in $M(2,n)$, which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$.

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

We fix an error on a $3$-cocycle in the original version of the paper ‘Endoscopy for Hecke categories, character sheaves and representations’. We give the corrected statements of the main results.

Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$. We show that there is a normal subgroup $T\leq G$ and a subgroup $B\leq K$ such that the indices $[G:T]$ and $[K:B]$ and the order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where $K=G$. We deduce a number of corollaries of this result. A typical application is that if $K$ 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 $\epsilon$-bounded. In the same spirit we consider the cases where $K$ is a term of the lower central series of $G$, or a Sylow subgroup, etc.

We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

Let G be a finite solvable group and let p be a prime divisor of $|G|$. We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.