This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$ -group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$ -adic vector space associated with  $M$ . This leads to our second variation, which is a study of the finite linear groups that can arise.

Let $UY_{n}(q)$ be a Sylow $p$-subgroup of an untwisted Chevalley group $Y_{n}(q)$ of rank $n$ defined over $\mathbb{F}_{q}$ where $q$ is a power of a prime $p$. We partition the set $\text{Irr}(UY_{n}(q))$ of irreducible characters of $UY_{n}(q)$ into families indexed by antichains of positive roots of the root system of type $Y_{n}$. We focus our attention on the families of characters of $UY_{n}(q)$ which are indexed by antichains of length $1$. Then for each positive root $\unicode[STIX]{x1D6FC}$ we establish a one-to-one correspondence between the minimal degree members of the family indexed by $\unicode[STIX]{x1D6FC}$ and the linear characters of a certain subquotient $\overline{T}_{\unicode[STIX]{x1D6FC}}$ of $UY_{n}(q)$. For $Y_{n}=A_{n}$ our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of $\text{Irr}(UE_{i}(q))$, $6\leqslant i\leqslant 8$, and $\text{Irr}(UF_{4}(q))$.

The generating graph $\unicode[STIX]{x1D6E4}(H)$ of a finite group $H$ is the graph defined on the elements of $H$, with an edge between two vertices if and only if they generate $H$. We show that if $H$ is a sufficiently large simple group with $\unicode[STIX]{x1D6E4}(G)\cong \unicode[STIX]{x1D6E4}(H)$ for a finite group $G$, then $G\cong H$. We also prove that the generating graph of a symmetric group determines the group.

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

Let $G$ be a finite group acting transitively on a set $\unicode[STIX]{x1D6FA}$ . We study what it means for this action to be quasirandom, thereby generalizing Gowers’ study of quasirandomness in groups. We connect this notion of quasirandomness to an upper bound for the convolution of functions associated with the action of $G$ on $\unicode[STIX]{x1D6FA}$ . This convolution bound allows us to give sufficient conditions such that sets $S\subseteq G$ and $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2}\subseteq \unicode[STIX]{x1D6FA}$ contain elements $s\in S,\unicode[STIX]{x1D714}_{1}\in \unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D714}_{2}\in \unicode[STIX]{x1D6E5}_{2}$ such that $s(\unicode[STIX]{x1D714}_{1})=\unicode[STIX]{x1D714}_{2}$ . Other consequences include an analogue of ‘the Gowers trick’ of Nikolov and Pyber for general group actions, a sum-product type theorem for large subsets of a finite field, as well as applications to expanders and to the study of the diameter and width of a finite simple group.

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points provide satisfactory convergence. Even for $g=2$, the classical approach using moment statistics requires about $2^{30}$ sample points to obtain such information.

In this paper, we obtain some criteria for $p$-nilpotency and $p$-supersolvability of a finite group and extend some known results concerning weakly $S$-permutably embedded subgroups. In particular, we generalise the main results of Zhang et al. [‘Sylow normalizers and $p$-nilpotence of finite groups’, Comm. Algebra43(3) (2015), 1354–1363].

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$, we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$.

The trace (or zeroth Hochschild homology) of Khovanov’s Heisenberg category is identified with a quotient of the algebra $W_{1+\infty }$. This induces an action of $W_{1+\infty }$ on the center of the categorified Fock space representation, which can be identified with the action of $W_{1+\infty }$ on symmetric functions.

In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ . He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G<\text{GL}_{n}(\mathbb{C})$ , $G$ has a semi-invariant of degree at most $Cn$ . This conjecture would imply that the ${\it\alpha}$ -invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$ , as introduced by Tian in 1987, is at most $C$ . We prove Thompson’s conjecture in this paper.

Suppose that a finite group G admits an automorphism of order 2n such that the fixed-point subgroup of the involution is nilpotent of class c. Let m = ) be the number of fixed points of . It is proved that G has a characteristic soluble subgroup of derived length bounded in terms of n, c whose index is bounded in terms of m, n, c. A similar result is also proved for Lie rings.

In this paper, we investigate the influence of certain subgroups of fixed prime power order on the $p$-supersolubility of finite groups.

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_{q}$. For this, it is essential to treat all the pure inner $\mathbb{F}_{q}$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the ‘trace of Frobenius’ functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko and Drinfeld’s theory of character sheaves on neutrally unipotent groups are in fact positive integral, confirming a conjecture due to Drinfeld.

Let $G$ be a finite group and $\mathsf{cd}(G)$ denote the set of complex irreducible character degrees of $G$. We prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is a sporadic simple group $H_{0}$ and such that $\mathsf{cd}(G)=\mathsf{cd}(H)$, then $G^{\prime }\cong H_{0}$ and there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert’s conjecture, we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$, so that we cannot extend the conjecture to almost simple groups.

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.

We introduce the concept of infinite cochain sequences and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) and also how they can be applied to prove that almost all groups in such a family have equivalent Quillen categories. We also include some examples of infinite families of $p$-groups from different coclass families that have equivalent Quillen categories.

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$.