Saturated fusion systems are categories modeling properties of conjugacy of p-elements in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group G generate a subnormal subgroup of G. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a local Langlands correspondence for all such representations. We obtain it as a specialization from a categorical version: an equivalence between the category of finite-length non-singular depth-zero G-representations and the category of finite-length right modules of a direct sum of twisted affine Hecke algebras constructed from Langlands parameters. We also show that our LLC and our equivalence of categories have several nice properties, for example, compatibility with parabolic induction and with twists by depth-zero characters.

In this paper, we prove a property of kernels of Brauer characters. We propose a candidate for the kernels of Isaacs’ partial characters, and we show that this candidate has the same property.

A subgroup R of a finite group G is called weakly subnormal in G if R is not subnormal in G but it is subnormal in every proper overgroup of R in G. In this paper, weak subnormality is used to construct a subgroup lattice of a finite soluble group containing the lattice of all subnormal subgroups. A new characterisation of Schmidt groups is also obtained: they are exactly those groups with all subgroups subnormal or weakly subnormal.

It is established that the sporadic simple groups, the alternating groups and the projective special linear groups $L_2(p)$ for primes $p \geq 5$ are uniquely determined by their group orders and the minimum nontrivial codegrees of their irreducible characters.

The depth of a subgroup H of a finite group G is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb {C}H \subseteq \mathbb {C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

Let $\lambda (G)$ be the maximum number of subgroups in an irredundant cover of the finite group G. We establish bounds on the order, exponent and derived length of the group in terms of this invariant.

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here, we verify some of these conjectures for fusion systems on an extraspecial group of order $p^3$, which contain among them the Ruiz–Viruel exotic fusion systems at the prime $7$. As a byproduct, we verify Robinson’s ordinary weight conjecture for principal p-blocks of almost simple groups G realizing such (nonconstrained) fusion systems.

For a group G and $m\ge 1$, $G^m$ denotes the subgroup generated by the elements $g^m$, where g runs through G. The subgroups not of the form $G^m$ are called nonpower subgroups. We extend the classification of groups with few nonpower subgroups from groups with at most nine nonpower subgroups to groups with at most 13 nonpower subgroups.

Let K be a genus one two-bridge knot. Let p be a prime number and let ${\mathbb {Z}}_{p}$ denote the ring of p-adic integers. In the spirit of arithmetic topology, we observe that if $p\neq 2$ and p divides (or $p=2$ and $2^3$ divides) the size of the 1st homology group of some odd-th cyclic branched cover of the knot K, then its group $\pi _1(S^3-K)$ admits a liminal $\mathrm { SL}_2{\mathbb {Z}}_p$-character. In addition, we discuss the existence of liminal $\mathrm {SL}_2{\mathbb {Z}}_{p}$-representations and give a remark on a general two-bridge knot. In the course of the argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

Given a group G and an automorphism $\varphi $ of G, two elements $x,y\in G$ are said to be $\varphi $-conjugate if $x=gy\varphi (g)^{-1}$ for some $g\in G$. The number $R(\varphi )$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi $ and the set $\{R(\varphi ) \mid \varphi \in \text {Aut}(G)\}$ is called the Reidemeister spectrum of G. We determine the Reidemeister spectrum of ZM-groups, extending some results of Senden [‘The Reidemeister spectrum of split metacyclic groups’, Preprint, 2022, arXiv:2109.12892].

Let ${\mathbb {F}_q}$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod p representations of ${G_q} = \text {GL}_2({{\mathbb {F}_q}})$ to ${G_p} = \text {GL}_2({\mathbb {F}_p})$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup ${B_q}$. We then analyze the restrictions of inductions from an anisotropic torus ${T_q}$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of f. The proofs depend on writing down explicit orbit decompositions of ${G_p} \backslash {G_q} / H,$ where $H = {B_q}$ or ${T_q}$ using the fact that ${G_q}/H$ is an explicit orbit in a certain projective line, along with Mackey theory.

Let G be a finite group and p be a prime number. An element g of G is called an $\mathrm {SM}^*$-vanishing element of G if there exists a strongly monolithic character $\chi $ of G satisfying $Z(\chi )=\ker \chi $ and $\chi (g)=0$. In this paper, we present some results on the relationship between the orders of $\mathrm {SM}^*$-vanishing elements of G and the structure of G.

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.

For a prime p, let $\mathcal {N}_p(G)$ denote the intersection of the normalisers of all non-p-nilpotent subgroups of a finite group G and set $\mathcal {N}_p(G)=G$ if G itself is p-nilpotent. We give some properties of $\mathcal {N}_p(G)$ and investigate the influence of $\mathcal {N}_p(G)$ on G.

In this article, we provide a specific characterization of invariants of classical Lie superalgebras from the super-analog of the Schur–Weyl duality in a unified way. We establish $\mathfrak {g}$-invariants of the tensor algebra $T(\mathfrak {g})$, the supersymmetric algebra $S(\mathfrak {g})$, and the universal enveloping algebra $\mathrm {U}(\mathfrak {g})$ of a classical Lie superalgebra $\mathfrak {g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak {g})^{\mathfrak {g}}$ of the projection from $T(\mathfrak {g})$ to $\mathrm {U}(\mathfrak {g})$ is surjective, which enables us to determine the generators of the center $\mathcal {Z}(\mathfrak {g})$ except for $\mathfrak {g}=\mathfrak {osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal {Z}(\mathfrak {p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial t-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte T-designs. We explore when a T-design is also a $T'$-design, where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

Commutator blueprints can be seen as blueprints for constructing RGD systems over $\mathbb {F}_2$ with prescribed commutation relations. In this paper, we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Also applying another result of the author, we obtain new examples of exotic RGD systems of universal type over $\mathbb {F}_2$. In particular, we generalize Tits’ construction of uncountably many trivalent Moufang twin trees to higher rank, we obtain an example of an RGD system of rank $3$ such that the nilpotency degree of the groups $U_w$ is unbounded, and we construct a commutator blueprint of type $(4, 4, 4)$ that is used to answer a question of Tits from the late $1980$s about twin buildings.

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R. The focus is on lattices, namely, finitely generated G-modules that are projective as R-modules, and on the full subcategory of all G-modules projective over R generated by the lattices. The stable category of such G-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.