Let $G$ be a finite group and let $\tau = (H_1, \ldots, H_t)$ be a $t$-tuple of core-free subgroups of $G$. We say that $\tau$ is regular if $G$ contains elements $g_1, \ldots, g_t$ such that $\bigcap_i H_i^{g_i} = 1$, which is equivalent to the existence of a regular $G$-orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_t$. Regular tuples were first investigated by Anagnostopoulou-Merkouri and Burness in a paper from 2024, partly motivated by the aim of seeking a natural generalisation of the classical and widely studied concept of a base for a transitive permutation group, which aligns with the special case where the $H_i$ are pairwise conjugate subgroups. In this paper, we focus on the case where $G$ is a finite almost simple classical group and each $H_i$ is a maximal subgroup contained in Aschbacher's collection $\mathcal{S}$ of irreducibly embedded subgroups. Our main theorem determines all the non-regular $t$-tuples of this form with $t \geqslant 2$, which extends earlier work by Burness, Guralnick and Saxl in the base size setting. In particular, we deduce that every pair of maximal subgroups in $\mathcal{S}$ is regular if $n \geqslant 15$, where $n$ is the dimension of the natural module for the socle of $G$, and this lower bound is best possible.
Let $d(G)$ denote the least size of a generating set of a finite group $G$. We prove that if $G$ has a family $\mathcal H$ of subgroups such that $d(H)\leq d$ for every $H\in\mathcal H$ and $\gcd\{\lvert G:H\rvert:H\in\mathcal H\}=1$, then $d(G)\leq d+1$. This gives an affirmative answer to Kourovka Problem 21.87. The proof reduces a minimal counterexample to a critical crown-based power with nonabelian socle. An exact crown multiplicity formula and a uniform lower bound for conditional generation give a lower bound for the number of crown factors. A subgroup containing a Sylow $2$-subgroup gives the contradictory upper bound, via a pointwise centralizer estimate for Sylow $2$-subgroups of finite simple groups.
Consider $V_*$, the subgroup of R. Thompson's group $V$ which stabilises $0^\omega$ under the natural action upon the Cantor set, $\{0, 1\}^\omega$. Let $G_*$ be any subgroup of a finitely generated group $G$. We show that $G_*$ is a context-free subgroup of $G$ if and only if $G_*$ is a pullback of $V_*$ under a homomorphism $G \rightarrow V$. In particular, this shows the existence of a hardest context-free membership problem. As a consequence, we prove that a group $G$ embeds into $V$ if and only if it is the transition group of a finite union of context-free automata, or equivalently, if there exist finitely many context-free subgroups of $G$ whose cores intersect trivially.
For a finite graph $G$ on $n$ vertices, let $\eta(G)$ denote the least order of a finite abelian group $\Gamma$ for which $G$ is an induced subgraph of some Cayley graph of $\Gamma$. Babai and S\'os (1985) settled the worst-case order of magnitude: it is $\Theta(n^2)$. We treat $\eta$ instead as an invariant of the individual graph, minimised over all finite abelian groups rather than over the cyclic groups alone, which is the restriction implicit in the literature on representation numbers modulo $n$. We prove a local order floor: $\eta(G)$ is at least the maximum of $n$ and twice the largest independence number of a neighbourhood of $G$. This localises at an arbitrary vertex the correspondence of Babai and S\'os between induced stars and sum-free sets; a corollary of the classification of maximum sum-free sets in abelian groups does not lower this floor, but restricts which host orders are admissible and so prunes the search. We determine $\eta$ exactly for paths, where it equals $n+1$, and for complete bipartite graphs $K_{a,b}$, where it equals $2\max(a,b)$ and meets the floor. A Cartesian product bound gives $\eta(P_m \,\square\, P_m) = (1+o(1))n$. We report certified exact values of $\eta$ for $22$ graphs, computed over all abelian groups. Seventeen of the $22$ optimal hosts are cyclic, so on most of these graphs the cyclic restriction costs nothing; where it bites, however, it is expensive. A search restricted to cyclic groups returns $36$ for the Petersen graph against the true value $16$, and $59$ for the Frucht graph against $27$. The cost of the restriction is concentrated rather than diffuse, and we identify the graphs on which it is paid. We also determine $\eta$ exactly for the double stars $D_{q,q}$ with $2 \le q \le 6$, obtaining $5q$ in each case. Since $\eta(D_{6,6}) = 30$ exceeds $2n = 28$, no constant below $15/7$ can bound $\eta(T)/n$ over all trees.
We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-covolume discrete subgroups that act ergodically on the Furstenberg boundary of $G$, providing a counterexample to a conjecture of Margulis for $G = SO(n,2), n>2$. The examples arise from lattices $\Gamma<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$ can fail to be $C^0$-locally rigid.
S. Dey, Mikołaj Frączyk, Sebastian Hurtado· 0 citations
Let G⩽Sym(Ω)$G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser H$H$ . A base for G$G$ is a subset of Ω$\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of G$G$ , denoted by b(G,Ω)$b(G, \Omega)$ . Equivalently, b(G,Ω)$b(G, \Omega)$ is the minimal positive integer k$k$ such that G$G$ has a regular orbit on the Cartesian product Ωk$\Omega ^k$ . A well‐known conjecture of Cameron from the 1990s asserts that if G$G$ is an almost simple primitive group and H$H$ is a so‐called non‐standard subgroup, then b(G,Ω)⩽7$b(G, \Omega) \leqslant 7$ , with equality if and only if G$G$ is the Mathieu group M24${\rm M}_{24}$ in its natural action of degree 24. This conjecture was settled in a series of papers by Burness et al.
Marina Anagnostopoulou-Merkouri· Journal of the London Mathem...· 0 citations
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\ge\nu(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.
Baoyu Zhang· 0 citations
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