We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

The first author [J. Brough, ‘On vanishing criteria that control finite group structure’, J. Algebra458 (2016), 207–215] has shown that for certain arithmetical results on conjugacy class sizes it is enough to consider only the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.

A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed “standard basis” through a triangular base change matrix with polynomial entries whose constant terms equal the identity matrix. Among the better known examples of canonical bases are the Kazhdan–Lusztig basis of Iwahori–Hecke algebras (see Kazhdan and Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184), Lusztig’s canonical basis of quantum groups (see Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3(2) (1990), 447–498) and the Howlett–Yin basis of induced $W$-graph modules (see Howlett and Yin, Inducing W-graphs I, Math. Z. 244(2) (2003), 415–431; Inducing W-graphs II, Manuscripta Math. 115(4) (2004), 495–511). This paper has two major theoretical goals: first to show that having bases is superfluous in the sense that canonicalization can be generalized to nonfree modules. This construction is functorial in the appropriate sense. The second goal is to show that Howlett–Yin induction of $W$-graphs is well-behaved a functor between module categories of $W$-graph algebras that satisfies various properties one hopes for when a functor is called “induction,” for example transitivity and a Mackey theorem.

We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the $q$-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for $q$-Schur superalgebras. We also construct infinitesimal and little $q$-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

Assume that $G$ is a finite group and $H$ is a 2-nilpotent Sylow tower Hall subgroup of $G$ such that if $x$ and $y$ are $G$-conjugate elements of $H\cap G^{\prime }$ of prime order or order 4, then $x$ and $y$ are $H$-conjugate. We prove that there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N=1$.

Let p ≥ 3 be a prime. A generalized multi-edge spinal group

$$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$

$b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$

In this paper, we introduce the Newton decomposition on a connected reductive $p$-adic group $G$. Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on $G$ and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori–Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe’s conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description of the structure of the rigid cocenter.

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

Let $G$ be a group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup $A$ of $G$ is $\unicode[STIX]{x1D70E}$-subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\unicode[STIX]{x1D70E}_{j}$-group for some $j=j(i)$ for $i=1,\ldots ,m$, and it is modular in $G$ if $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when $X\leq Z\leq G$ and $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when $Y\leq G$ and $A\leq Z\leq G$. The group $G$ is called $\unicode[STIX]{x1D70E}$-soluble if every chief factor $H/K$ of $G$ is a finite $\unicode[STIX]{x1D70E}_{i}$-group for some $i=i(H/K)$. In this paper, we describe finite $\unicode[STIX]{x1D70E}$-soluble groups in which every $\unicode[STIX]{x1D70E}$-subnormal subgroup is modular.

Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

Given an odd prime $p$, we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow $p$-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is $3$, we prove that simple modules in the principal block all lie at the end of their components.

We derive some structural properties of a trifactorised finite group $G=AB=AC=BC$, where $A$, $B$, and $C$ are subgroups of $G$, provided that $A=A_{\unicode[STIX]{x1D70B}}\times A_{\unicode[STIX]{x1D70B}^{\prime }}$ and $B=B_{\unicode[STIX]{x1D70B}}\times B_{\unicode[STIX]{x1D70B}^{\prime }}$ are $\unicode[STIX]{x1D70B}$-decomposable groups, for a set of primes $\unicode[STIX]{x1D70B}$.

For a group G, a weak Cayley table isomorphism is a bijection f : G → G such that f(g1g2) is conjugate to f(g1)f(g2) for all g1, g2 ∈ G. The set of all weak Cayley table isomorphisms forms a group (G) that is the group of symmetries of the weak Cayley table of G. We determine (G) for each of the 17 wallpaper groups G, and for some other crystallographic groups.

For an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every x ∈ G all sufficiently long commutators [. . .[[x, g], g], . . ., g] belong to ${\mathscr E}$(g). A finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group G every element has an Engel sink generating a subgroup of rank r, then G has a normal subgroup N of rank bounded in terms of r such that G/N is nilpotent.

Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^{r}-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

Let $G$ be a finite solvable group and let $p$ be a prime. We prove that the intersection of the kernels of irreducible monomial $p$-Brauer characters of $G$ with degrees divisible by $p$ is $p$-closed.