Let $\Gamma $ be a group and ${\Gamma }^{\prime } $ be a subgroup of $\Gamma $ of finite index. Let $M$ be a $\Gamma $-module. It is shown that $M$ is (strongly) Gorenstein flat if and only if it is (strongly) Gorenstein flat as a ${\Gamma }^{\prime } $-module. We also provide some criteria in which the classes of Gorenstein projective and strongly Gorenstein flat $\Gamma $-modules are the same.

We prove an analogue of Koszul duality for category $ \mathcal{O} $ of a reductive group $G$ in positive characteristic $\ell $ larger than $1$ plus the number of roots of $G$. However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus $2$.

Let $\mathfrak{F}$ be a locally compact nonarchimedean field with residue characteristic $p$, and let $\mathrm{G} $ be the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. For $k$ an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke $k$-algebra ${\mathrm{H} }^{\prime } $ and of the pro-$p$ Iwahori–Hecke $k$-algebra $\mathrm{H} $ of $\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of $\mathrm{G} $. If $\mathrm{G} $ is semisimple, we also show that this upper bound is sharp, that both $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of $\mathrm{H} $ (respectively ${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $.

When $k$ has characteristic $p$, we prove that in ‘most’ cases $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ have infinite global dimension. In particular, we deduce that the category of smooth $k$-representations of $\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$ generated by their invariant vectors under the pro-$p$ Iwahori subgroup has infinite global dimension (at least if $k$ is algebraically closed).

If $X$ is a subgroup of a group $G$, the cardinal number $\min \{ \vert X: X_{G}\vert , \vert {X}^{G} : X\vert \} $ is called the normal oscillation of $X$ in $G$. It is proved that if all subgroups of a locally finite group $G$ have finite normal oscillation, then $G$ contains a nilpotent subgroup of finite index.

In this paper we investigate some subclasses of strongly regular congruences on an $E$-inversive semigroup $S$. We describe the minimum and the maximum strongly orthodox congruences on $S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on $S$ with characteristic trace $\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an $E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $E$-supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G= HT$ and $H\cap T\leq {H}_{eG} $, where ${H}_{eG} $ denotes the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-quasinormally embedded in $G$. In this paper, some new characterisations of $p$-supersolubility of finite groups are given under the assumption that some primary subgroups are $E$-supplemented.

The so-called Burnside–Dixon–Schneider (BDS) method, currently used as the default method of computing character tables in GAP for groups which are not solvable, is often inefficient in dealing with groups with large centres. If $G$ is a finite group with centre $Z$ and $\lambda $ a linear character of $Z$, then we describe a method of computing the set $\mathrm{Irr} (G, \lambda )$ of irreducible characters $\chi $ of $G$ whose restriction ${\chi }_{Z} $ is a multiple of $\lambda $. This modification of the BDS method involves only $\vert \mathrm{Irr} (G, \lambda )\vert $ conjugacy classes of $G$ and so is relatively fast. A generalization of the method can be applied to computation of small sets of characters of groups with a solvable normal subgroup.

Supplementary materials are available with this article.

We show that there are only finitely many primes $p$ such that $\mathrm{PSL} (2, p)$ has a minimal generating set of size four.

Supplementary materials are available with this article.

We construct categorical braid group actions from 2-representations of a Heisenberg algebra. These actions are induced by certain complexes which generalize spherical (Seidel–Thomas) twists and are reminiscent of the Rickard complexes defined by Chuang–Rouquier. Conjecturally, one can relate our complexes to Rickard complexes using categorical vertex operators.

We show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

A semiring is a set $S$ with two binary operations $+ $ and $\cdot $ such that both the additive reduct ${S}_{+ } $ and the multiplicative reduct ${S}_{\bullet } $ are semigroups which satisfy the distributive laws. If $R$ is a ring, then, following Chaptal [‘Anneaux dont le demi-groupe multiplicatif est inverse’, C. R. Acad. Sci. Paris Ser. A–B 262 (1966), 274–277], ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. In Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136], it is proved that if $R$ is a regular ring then ${R}_{\bullet } $ is orthodox if and only if ${R}_{\bullet } $ is a union of groups if and only if ${R}_{\bullet } $ is an inverse semigroup if and only if ${R}_{\bullet } $ is a Clifford semigroup. The latter result, also known as Zeleznikow’s theorem, does not hold in general even for semirings $S$ with ${S}_{+ } $ a semilattice Zeleznikow [‘Regular semirings’, Semigroup Forum 23 (1981), 119–136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow’s theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings $S$ for which ${S}_{+ } $ is a semilattice.

We study the mixing properties of progressions $(x, xg, x{g}^{2} )$, $(x, xg, x{g}^{2} , x{g}^{3} )$ of length three and four in a model class of finite nonabelian groups, namely the special linear groups ${\mathrm{SL} }_{d} (F)$ over a finite field $F$, with $d$ bounded. For length three progressions $(x, xg, x{g}^{2} )$, we establish a strong mixing property (with an error term that decays polynomially in the order $\vert F\vert $ of $F$), which among other things counts the number of such progressions in any given dense subset $A$ of ${\mathrm{SL} }_{d} (F)$, answering a question of Gowers for this class of groups. For length four progressions $(x, xg, x{g}^{2} , x{g}^{3} )$, we establish a partial result in the $d= 2$ case if the shift $g$ is restricted to be diagonalizable over $F$, although in this case we do not recover polynomial bounds in the error term. Our methods include the use of the Cauchy–Schwarz inequality, the abelian Fourier transform, the Lang–Weil bound for the number of points in an algebraic variety over a finite field, some algebraic geometry, and (in the case of length four progressions) the multidimensional Szemerédi theorem.

Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabiliser is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss conjecture.

In the 2006 edition of the Kourovka Notebook, Berkovich poses the following problem (Problem 16.13): Let $p$ be a prime and $P$ be a finite $p$-group. Can $P$ have every maximal subgroup special? We show that the structure of such groups is very restricted, but for all primes there are groups of arbitrarily large size with this property.

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [E. Cline, B. Parshall, and L. Scott, Cohomology of finite groups of Lie type, I, Publ. Math. Inst. Hautes Études Sci. 45 (1975), 169–191] and [E. Cline, B. Parshall, L. Scott and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143–163]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we show that, for sufficiently large $r$, depending only on the root system and $m$, and not on the prime $p$ or the irreducible module $L$, there are isomorphisms ${\mathrm{H} }^{m} (G({p}^{r} ), L)\cong {\mathrm{H} }^{m} (G({p}^{r} ), {L}^{\prime } )\cong { \mathrm{H} }_{\mathrm{gen} }^{m} (G, {L}^{\prime } )\cong {\mathrm{H} }^{m} (G, {L}^{\prime } )$, where the subscript ‘gen’ refers to generic cohomology and ${L}^{\prime } $ is a constructibly determined irreducible ‘shift’ of the (arbitrary) irreducible module $L$ for the finite Chevalley group $G({p}^{r} )$. By a famous theorem of Steinberg, both $L$ and ${L}^{\prime } $ extend to irreducible modules for the ambient algebraic group $G$ with ${p}^{r} $-restricted highest weights. This leads to the notion of a module or weight being ‘shifted $m$-generic’, and thus to the title of this paper. Our approach is based on questions raised by the third author in [D. I. Stewart, The second cohomology of simple ${\mathrm{SL} }_{3} $-modules, Comm. Algebra 40 (2012), 4702–4716], which we answer here in the cohomology cases. We obtain many additional results, often with formulations in the more general context of ${ \mathrm{Ext} }_{G({p}^{r} )}^{m} $ with irreducible coefficients.

A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.

In this paper we introduce the notion of finite virtual length for profinite groups (that is, every series has a bounded number of infinite factors) and we prove a Jordan–Hölder type theorem for profinite groups with finite virtual length. More structural results are provided in the pronilpotent and $p$-adic analytic cases.

All groups considered in this paper are finite. A subgroup $H$ of a group $G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of $G$ containing $H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of $G$ has index a power of a prime if and only if $G/ \Phi (G)$ is a solvable PST-group. Let $\mathfrak{X}$ denote the class of groups $G$ all of whose primitive subgroups have prime power index. It is established here that a group $G$ is a solvable PST-group if and only if every subgroup of $G$ is an $\mathfrak{X}$-group.