Papers

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

We give results on the question of whether the intersection numbers of a primitive group association scheme determine it up to combinatorial isomorphism. For $G=\operatorname{PSL}(2,q)$, where $q$ is an odd prime power with $q=11$ or $q\ge 17$, or $q=2^f$ with $f\ge3$, we construct a Schur partition that is algebraically isomorphic to the partition of $G$ into conjugacy classes but not combinatorially isomorphic to it. Consequently, the corresponding primitive group association schemes are not determined up to combinatorial isomorphism by their intersection numbers; in particular, they are non-separable. For $\mathfrak A_6$ and $\mathfrak A_8$, we also explicitly construct Schur partitions that are algebraically isomorphic to the corresponding partitions into conjugacy classes but not combinatorially isomorphic to them.

There is a conjecture, usually attributed to J. G. Thompson, that a finite simple nonabelian group $G$ has a conjugacy class $\Psi$ of covering number 2; i.e $G = \Psi^2$. We show that the commutator subgroup $\Omega(V,Q)$ of the orthogonal group $O(V,Q)$ of a hyperbolic quadratic vector space $(V, Q)$ over a field $K$ has a conjugacy class $\Psi$ such that $\Omega(V,Q) = \Psi^2 \cup -\Psi^2$.

Let $G$ be a finite group. For every subgroup $H\leq G$, let $\pi_H$ be the permutation character of the action of $G$ on the left cosets of $H$. We prove that the characters $\pi_H$, with $H$ running through representatives of the conjugacy classes of subgroups of $G$, are linearly independent if and only if $G$ is cyclic. In particular, no finite insoluble group has the property asked for in Kourovka Notebook Problem~11.9. The proof uses only the fixed-point formula for a coset action and an elementary triangular-matrix argument.

We study unordered triples of order-$n$ floretion base vectors whose tile centroids form nondegenerate equilateral triangles. A scaled integer centroid map turns Euclidean completion into exact arithmetic on a triangular lattice, and a residue obstruction modulo $3$ shows that every equilateral centroid triangle uses three tiles of one orientation. Combined with finite triangular-lattice completion counts, this gives $|E_n|=4^n(4^n-1)/12$. Equilateral triangles containing the identity word $e_n$ as a vertex form an entirely nonlocal family of cardinality $(4^n+(-2)^n-2)/3$. For synchronized local $\gamma$-cycles, $|L_n|=(7^n-4^n)/3$ and $|L_n|/|E_n|\sim4(7/16)^n$, while on the no-$e$ support $S_n=\{i,j,k\}^n$ locality is exhaustive and $|E_n^S|=|L_n^S|=(2^n-1)3^{n-1}$. The union of the three main axes supports exactly $|E_n^{\rm ax}|=4^{n-1}+2^n-2$ equilateral triangles, split into the branches $x=y=z$ and $x+y+z=0$. For $T\in E_n$, the unsigned vertex product defines a product point $C_T$; a digitwise parity criterion characterizes $C_T=Q_T$ on local cycles and yields Fibonacci subfamilies. Multiplication-generation is equivalent to $p(T)=e_n$, hence $C_T=0$; locally this gives exactly the nontrivial global $\gamma$-orbits, and exact enumeration through order $6$ finds no nonlocal example. Retaining the signs discarded by the unsigned product gives a second classifier: a triangle has scalar vertex-sum square exactly when its three vertices pairwise anticommute. For local cycles this occurs exactly when $|S|$ is odd, giving $|\mathrm{AC}_n\cap L_n|=(7^n-1)/6$, while nonlocal pairwise-anticommuting examples already occur in order $3$.

An $n$-tuple of positive integers is $G$-harmonic if there are subgroups of $G$ having those indices whose cosets can be chosen pairwise disjoint, and $\mathbb{Z}$-harmonic if there are pairwise disjoint residue classes with those moduli. Ginosar asked whether every $G$-harmonic tuple is $\mathbb{Z}$-harmonic. Margolis and Schnabel proved this for tuples of length at most $4$, and analysed a particular family of length-$5$ tuples that would yield a counterexample if any member were $G$-harmonic. We show that the bound $4$ is sharp: $(6,6,6,10,15)$ is $A_5$-harmonic but not $\mathbb{Z}$-harmonic. Moreover, the five pairwise disjoint cosets realising this tuple can be extended to a coset partition of $A_5$ using only cosets of indices $6$, $10$, and $15$. The index tuple of this partition is not $\mathbb{Z}$-harmonic; because its indices repeat, this does not contradict the Herzog–Schönheim conjecture. We also prove that no member of the length-$5$ family analysed by Margolis and Schnabel in connection with possible counterexamples is $G$-harmonic for any group $G$.

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn–Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.

For any group $G$, the set of all nonempty subsets of $G$ forms an additively idempotent semiring under set-theoretic union and elementwise multiplication, called the power semiring of $G$ and denoted by $\mathcal{P}(G)$. We prove that for a finite group $G$, $\mathcal{P}(G)$ has no finite basis for its identities if and only if $|G| \geq 3$. This completes the classification of the power semirings of finite groups with respect to the finite basis property.

Let $H_1, H_2$ be locally quasiconvex, torsion-free subgroups of a hyperbolic group $G$. We prove $\mathrm{rank}(H_1 \cap H_2)$ can be bounded in terms of $\mathrm{rank}(H_1)$ and $\mathrm{rank}(H_2)$.

We are interested in the following conjecture of Wilson from 1981: every locally nilpotent countably categorical group is nilpotent. Following our previous work on the Lie algebra analogue of the conjecture, we use Lie methods to deduce the first nontrivial cases of the Wilson conjecture for n-Engel groups: countably categorical 3-Engel groups and 4-Engel 5-groups are nilpotent, from which we also conclude that countably categorical 4-Engel groups of odd exponent are nilpotent. This is implemented via an exceptional case of the Lazard correspondence, checked using computer algebra systems. We also study the transfer of nilpotency results between the three categories: groups, Lie algebras, associative algebras. Among other things, we prove that the Wilson conjecture implies the analogous nilpotency statement for associative algebras (modulo the commutative case).

In this note, we prove a recent conjecture of Hung, Martínez and Navarro on the sums of character degrees of finite groups.

Let $F_n$ be a free group of rank $n\geq3$, freely generated by $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ denote the Nielsen automorphism of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $d_{i,j}(x_k)=x_k$ for $k\neq i$, and let $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$. We determine the lower central series of $D_n$. We first prove that, for $1\leq r<i\leq n$, $d_{i,r}\in \gamma_{i-r}(D_n)\setminus\gamma_{i-r+1}(D_n)$. For each $i=2,\ldots,n$, this calculation leads to a filtration $\{W_{i,c}\}_{c\geq1}$ of $U_i=\langle d_{i,1},\ldots,d_{i,i-1}\rangle$. For every $c\geq1$, we obtain an explicit iterated semidirect-product decomposition of $\gamma_c(D_n)$ in terms of the subgroups $W_{i,c}$, and prove that $U_i\cap\gamma_c(D_n)=W_{i,c}$ for $i=2,\ldots,n$. The construction also gives an explicit basis for each quotient $\gamma_c(D_n)/\gamma_{c+1}(D_n)$ in terms of basic commutators and determines the exact lower-central depth of every such commutator. Consequently, $D_n$ is a Magnus group.

We show that if M is a full II_infinity factor such that the fundamental group of a finite corner is R and Out(M) is a locally compact second countable group, then M admits either a unique continuous trace scaling flow or a continuum of pairwise distinct ones, up to cocycle conjugacy. Equivalently, there is either a unique III_1 factor or a continuum of pairwise non-isomorphic III_1 factors with continuous core M. We apply this dichotomy result to factors constructed in [Dep12] and [Cha25] where Out(M) is a Lie group. We also give two independent constructions of full factors admitting a continuum of pairwise distinct trace-scaling flows where the outer automorphism group is unknown.

The Brauer–Fowler theorem bounds the order of a finite simple group in terms of the order of the centralizer of an involution. Hartley proved an automorphism version, bounding the order of a finite simple group in terms of the order of an automorphism and the order of its fixed-point subgroup. Strunkov asked whether, in the involution case, the order of the centralizer can be replaced by the number of involutions commuting with the given involution; this was recently answered affirmatively by Skresanov. Motivated by this question, and in contrast with Hartley's use of the order of a fixed-point subgroup, we study what can be deduced from counting elements of prescribed prime order inside such a subgroup. We show that the direct analogue of Skresanov's result fails for elements of odd prime order. We prove that if \(G\) is a finite simple group, \(x\in G\) has odd prime order \(p\), \(C_G(x)\) contains at most \(k\) elements of order \(p\), and the exponent of \(C_G(x)\) is at most \(e\), then \(|G|\) is bounded in terms of \(k\) and \(e\).

We study the class of congruences on $F$-inverse monoids that respect the unary operation $a\mapsto\mx{a}$, assigning to each element the maximum element of its $\sigma$-class, which we refer to as $F$-congruences. We characterize kernel normal systems and congruence pairs on $F$-inverse monoids which give rise to such congruences and establish a number of further related results including a classification of $F$-congruence-free $F$-inverse monoids.

Let $G=F\Gamma$ be a factorization of a finite group, with neither factor assumed normal and with $F\cap\Gamma$ allowed to be nontrivial. We prove that $\exp(G)$ divides $\operatorname{lcm}(|F|,|\Gamma|)$, or equivalently that $\gcd([G:F],[G:\Gamma])\exp(G)$ divides $|G|$. This answers a cohomological divisibility question posed by Natale. Combining the group-theoretic divisibility with Natale's exponent bound and a lifting argument, we prove Kashina's exponent conjecture, in the arbitrary-field formulation of Etingof and Gelaki, for every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$ for which $\operatorname{Rep}(H\otimes_k\overline{k})$ is group-theoretical. The same argument proves the corresponding degree-three cohomological divisibility for coefficients in an arbitrary $G$-module. For complex group-theoretical categories, we also establish Frobenius-Schur exponent divisibility under a cohomological factorization hypothesis, without assuming a fiber functor. We derive applications to low-dimensional Hopf algebras and abelian extensions.

Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavskiĭ's result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.

We study conditions for a countable group acting on a connected locally finite hyperbolic graph to induce a hyperfinite orbit equivalence relation on the Gromov boundary of the graph in terms of tree-decompositions of the graph. We prove that for a connected locally finite hyperbolic graph $X$ equipped with an action of a countable group $G$, if $(T, \beta)$ is a $G$-invariant tree-decomposition of $X$ such that each bag induces a connected subgraph $X_t$ of $X$ for each $t \in V(T)$, each adhesion set is finite and such that there are only finitely many $G$-orbits of edges of $T$, then the orbit equivalence relation of $G$ acting on the Gromov boundary $\partial X$ is hyperfinite provided the orbit equivalence relation of $G$ acting on $\partial T$ is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on $\partial X_t$ are all hyperfinite. We show that the converse also holds if $(T, \beta)$ satisfies the additional property that each adhesion set distinguishes at least two ends of $X$.

A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.

In 1985, Odoni initiated the study of arboreal representations and the fixed-point proportion, motivated by prime density problems in arithmetic dynamics. Since then, many questions regarding the connection between the dynamics of rational functions and the Galois groups associated to their dynamics (iterated Galois groups) have been posed. In this article, we give several contributions to this connection with the introduction of the concept of virtually mixing groups. This new concept is related to the capability of self-replication of the iterated Galois groups, and when it fails to be completely self-replicated, the failure is always by a finite index normal subgroup. It turns out that the quotient by this subgroup contains valuable information of the properties of the rational function. First, we investigate the inverse Galois problem, showing that iterated Galois groups of rational functions are always virtually mixing and the failure (the quotient by this finite-index subgroup) is related to geometric properties of the rational function. As a consequence, we prove that the rational function is induced by an endomorphism of an algebraic curve (dynamical pullback) if and only if this failure is non-trivial. Finally, we solve the main open problem of the fixed-point proportion of geometric iterated Galois groups of rational functions, by showing that the fixed-point proportion is zero when the map is not a dynamical pullback. This result has direct applications to prime density problems, proportion of periodic points in the reduction of maps and proportion of periodic points over finite fields. The proofs in this article rely on a combination of techniques and results from group theory, ergodic theory, probability, number theory, complex dynamics, energy transport in graphs, arithmetic dynamics and algebraic geometry.

In this paper, we investigate the eigenvalues of character degree graphs, with particular emphasis on the arithmetic properties of their spectra. First, we study \((n-2)\)-regular character degree graphs of solvable groups and derive an explicit formula for their characteristic polynomials. We show that all their eigenvalues are rational and, consequently, that their splitting field is \(\mathbb{Q}\). We then consider supergraphs obtained by adding edges to these graphs and prove that the corresponding splitting field is a quadratic extension of \(\mathbb{Q}\). Next, using their structural decomposition, we examine a general class of Lewis graphs. For this class, we establish bounds on both the number of irrational eigenvalues and the degree of the associated splitting fields. Finally, we investigate prime character degree graphs of diameter \(3\), focusing on the arithmetic nature of their eigenvalues and the degree of their splitting fields.

Agol in his 2014 ICM proceedings article \cite[Question 11]{Agol14} asks whether braid groups are (virtually) RFRS. We answer this positively by showing that pure braid groups are RFRS. As a consequence, several families of Artin groups are virtually RFRS, including those of type $A_n$ (the braid groups), $B_n=C_n$, $\widetilde A_n$, and $\widetilde C_n$. Our results also provide evidence toward the problem of whether braid groups, and more generally Artin groups, are virtually special; see \cite[Problem 9.4]{HagWi10}, \cite[Problem 13.4]{Wise14}.

This article shows that the exact completion as a regular category of the homological category of Polish groups and continuous group homomorphisms is the category of groups with a Polish cover and Borel-definable group homomorphisms. We also obtain several equivalent characterizations for the morphisms in the exact completion. An analogous description is deduced for several important subcategories of the category of Polish groups.

We show that the FC-center of Thompson's group $F$, the subgroup consisting of all elements with finite conjugacy classes, is trivial.

We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The minimal synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.

The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero–nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.

In this short note, we address problems concerning the characterization of simple and related groups by various arithmetic invariants. We propose a uniform approach to such questions and discuss both known results and open problems in this area.

The size of minimal systems of generators and these systems themselves for groups $ESL_n \left[ \mathbb{Z} \right]$ and $ESL_n \left( \mathbb{F}_p \right)$ were found. Moreover we obtain triples of involutions with\textit{ two commuting involu\-tions} generating $ESL_5 \left[ \mathbb{Z} \right]$, $ESL_5 \left[ \mathbb{Z} \right]$ relatively, that are Mazurov triples.

We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid $S$ being LO (for every $s,t \in S$, we have that $s \in St$ or $t \in Ss$) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when $S$ is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when $S$ is LO.

Let \(k\geq 3\), \(n\geq 1\), and let \(H_{k,n}=\Sh(C_k,n)\) be the group generated by the standard \(k\) pile perfect shuffle and cyclic pile permutation on a deck of \(kn\) cards. We prove that \(H_{k,n}\) is \(2\)-transitive whenever \(n\) is not a power of \(k\). Residual commutators give translations supported on two pile labels, and a strongly connected digit digraph propagates these translations throughout the deck, a separate argument resolves the antipodal support case. We then combine this result with fixed point ratio bounds for primitive groups and explicit boundary calculations to determine \(H_{k,n}\) for all \(k\) and \(n\). If \(n=k^f\), then \(H_{k,n}\cong C_k\wr C_{f+1}\). If \(k=4\) and \(n=2\cdot4^j\), then \(H_{4,n}\cong\AGL(2j+3,2)\). In every other case, \(H_{k,n}\) is \(\Alt(kn)\) or \(\Sym(kn)\), according to the parity of its generators. This proves Conjecture~1.10 of Amarra, Morgan and Praeger and Conjecture~5.1 of Xia, Zhang, Zhang and Zhu. More generally, we classify \(\Sh(P,n)\) for every pile group \(P\) containing \(C_k\), and obtain the odd \(k\) part of their Conjecture~5.2.