For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed.

This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis.

For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$. We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all $n>1$.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$.

Given a finite group G, we denote by $L(G)$ the subgroup lattice of G and by ${\cal CD}(G)$ the Chermak–Delgado lattice of G. In this note, we determine the finite groups G such that $|{\cal CD}(G)|=|L(G)|-k$, for $k=1,2$.

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type $E_8$, $F_4$, and $G_2$. In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly $\textrm{SL}_{2n+1}(\mathbb{R})$, $\textrm{SL}_{2n+1}(\mathbb{C})$, $\textrm{SL}_n(\mathbb{H})$, or groups of type $E_6$.

We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, a concept recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion, we count the number of orbits for the action of the subgroup of fixed points (respectively periodic) points on the set of infinite fixed (respectively periodic) points. Finally, we study the dynamics of infinite points: for automorphisms and some endomorphisms, defined in a precise way, fitting a classification given by Delgado and Ventura, we prove that every infinite point is either periodic or wandering, which implies that the dynamics is asymptotically periodic.

We classify embeddings of the finite groups $A_4$, $S_4$ and $A_5$ in the Lie group $G_2(\mathbb C)$ up to conjugation.

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 give a classification of the maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analog of Cartan’s theorem that all maximal tori in a connected compact Lie group are conjugate.

A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if $\langle Q,g\rangle $ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that $G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that $G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

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.

We call a packing of hyperspheres in n dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the n-dimensional Euclidean space; and every sphere in the packing is a member of a cluster of $n+2$ mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The $E_7$, $E_8$ and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.

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.

In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models. One consequence of our results is that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE. Our techniques are based on invariant theory and castling transforms.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.

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 $

$\ell \geq 5$

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