We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$ . As a main step, we derive a labelling set for the irreducible $2$ -Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

We present several results on the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver.

Linckelmann and Murphy have classified the Morita equivalence classes of p-blocks of finite groups whose basic algebra has dimension at most $12$ . We extend their classification to dimension $13$ and $14$ . As predicted by Donovan’s conjecture, we obtain only finitely many such Morita equivalence classes.

A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups.

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

Given a finite group G, we denote by Δ(G) the graph whose vertices are the proper subgroups of G and in which two vertices H and K are joined by an edge if and only if G = ⟨H, K⟩. We prove that if there exists a finite nilpotent group X with Δ(G) ≅ Δ(X), then G is supersoluble.

Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

It is known that there exists a first-order sentence that holds in a finite group if and only if the group is soluble. Here it is shown that the corresponding statements with ‘solubility’ replaced by ‘nilpotence’ and ‘perfectness’, among others, are false.

These facts present difficulties for the study of pseudofinite groups. However, a very weak form of Frattini’s theorem on the nilpotence of the Frattini subgroup of a finite group is proved for pseudofinite groups.

For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$-torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$-representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$-projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200(1) (2015), 1–96].

Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$, with $\ell$ different from $p$. We define “nilpotent lifts” of irreducible generic $k$-representations of $GL_{n}(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$-representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

Let $G$ be a finite group and $p$ be an odd prime. We show that if $\mathbf{O}_{p}(G)=1$ and $p^{2}$ does not divide every irreducible $p$-Brauer character degree of $G$, then $|G|_{p}$ is bounded by $p^{3}$ when $p\geqslant 5$ or $p=3$ and $\mathsf{A}_{7}$ is not involved in $G$, and by $3^{4}$ if $p=3$ and $\mathsf{A}_{7}$ is involved in $G$.

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathscr{P}$ be the class of $p$-groups of order $p^{n}$ ($n\geq 3$). Consider the function $\unicode[STIX]{x1D6FC}:\mathscr{P}\longrightarrow (0,1]$ given by $\unicode[STIX]{x1D6FC}(G)=|C(G)|/|G|$. In this paper, we determine the second minimum value of $\unicode[STIX]{x1D6FC}$, as well as the corresponding minimum points. Since the problem of finding the second maximum value of $\unicode[STIX]{x1D6FC}$ has been solved for $p=2$, we focus on the case of odd primes in determining the second maximum.