Volume 28, Issue 3

Let 𝐺 be a group acting 2-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where 𝐺 has both P ⁢ Γ ⁢ L 3 ⁢ ( 4 ) \mathrm{P}\Gamma\mathrm{L}_{3}(4) and P ⁢ Γ ⁢ L 3 ⁢ ( 5 ) \mathrm{P}\Gamma\mathrm{L}_{3}(5) as local actions. We show that, for each half-tree T a T_{a} , the local action of the rigid stabilizer of T a T_{a} at the root of T a T_{a} contains the soluble residual of the point stabilizer of the local action of 𝐺. In particular, 𝐺 is locally prosoluble if and only if its local actions have soluble point stabilizers; if 𝐺 is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in 𝐺. These results are partly inspired by Radu’s classification of groups acting boundary-2-transitively on trees with local action containing the alternating group, and partly based on the author’s recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks.

A group 𝐺 is twisted conjugacy separable if, for every automorphism 𝜑, distinct 𝜑-twisted conjugacy classes can be separated in a finite quotient. Likewise, 𝐺 is completely twisted conjugacy separable if, for any group 𝐻 and any two homomorphisms φ , ψ \varphi,\psi from 𝐻 to 𝐺, distinct ( φ , ψ ) (\varphi,\psi) -twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that, for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite.

Given a topological 𝐺-space, we consider equations with constants over 𝐺. In particular, we formulate some very general conditions on words with constants w ⁢ ( y ̄ , g ̄ ) w(\bar{y},\bar{g}) over 𝐺 which guarantee that the inequality w ⁢ ( y ̄ , g ̄ ) ≠ 1 w(\bar{y},\bar{g})\neq 1 has a solution in 𝐺. These results are illustrated in some typical situations. In particular, standard actions of Thompson’s group 𝐹 and branch groups are considered.

For every integer n ≥ 2 n\geq 2 , we consider a family { π w } w ∈ { 0 , 1 , … , n − 1 } N \{\pi_{w}\}_{w\in\{0,1,\ldots,n-1\}^{\mathbb{N}}} of irreducible representations of the Cuntz algebra O n \mathcal{O}_{n} . All these representations (except one) are shown to be equivalent to those arising from the orbits of the interval map dynamical system ( I , f ) (I,f) with f ⁢ ( x ) = n ⁢ x ⁢ ( mod ⁢ 1 ) f(x)=nx\ (\mathrm{mod}\ 1) . We consider the embeddings V = V 2 ⊂ V n ⊂ O n V=V_{2}\subset V_{n}\subset\mathcal{O}_{n} of the Thompson group 𝑉 in the Higman–Thompson group V n V_{n} obtained by Birget and then from V n V_{n} into O n \mathcal{O}_{n} which was obtained independently by Birget and Nekrashevych. The restriction of π w \pi_{w} to V n V_{n} is still irreducible; however, the restriction of π w \pi_{w} to 𝑉 is no longer irreducible, and we obtain the corresponding irreducible decomposition in terms of quasi-regular representations.

Given a regular covering map φ : Λ → Γ \varphi\colon\Lambda\to\Gamma of graphs, we investigate the subgroup LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) of the automorphism group Aut ⁡ ( A Γ ) \operatorname{Aut}(A_{\Gamma}) of the right-angled Artin group A Γ A_{\Gamma} . This subgroup comprises all automorphisms that can be lifted to automorphisms of A Λ A_{\Lambda} . We first show that LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) is generated by a finite subset of Laurence’s elementary automorphisms. For the subgroup FAut ⁡ ( φ ) \operatorname{FAut}(\varphi) of Aut ⁡ ( A Λ ) \operatorname{Aut}(A_{\Lambda}) that consists of lifts of automorphisms in LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) , there exists a natural homomorphism FAut ⁡ ( φ ) → LAut ⁡ ( φ ) \operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi) induced by 𝜑. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup IA ⁡ ( A Λ ) \operatorname{IA}(A_{\Lambda}) and deduce a short exact sequence reminiscent of results from the Birman–Hilden theory for surfaces.

Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ K = K ⁢ H HK=KH for all subgroups 𝐾 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is said to be a PT \mathrm{PT} -group (or to have the PT \mathrm{PT} -property) if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, 𝐻 is permutable in 𝐺. The aim of this paper is to investigate the behaviour of uncountable soluble groups of cardinality ℵ whose proper subgroups of cardinality ℵ have the PT \mathrm{PT} -property. We prove that such a group is a PT \mathrm{PT} -group and so are all its subgroups.

Semiabelian groups are a family of finite solvable groups that have played an important role in the inverse Galois problem and the minimal ramification problem. In this paper, we establish several sufficient conditions for a solvable group to be semiabelian mainly by using its Sylow subgroups. We also present a family of non-semiabelian 2-groups.

In a graph Γ = ( V , E ) \Gamma=(V,E) , we consider the common closed neighbourhood of a subset of vertices and use this notion to introduce a Moore closure operator in 𝑉. We also consider the closed twin equivalence relation in which two vertices are equivalent if they have the same closed neighbourhood. Those notions are explored deeply in the case when Γ is the power graph associated with a finite group 𝐺. In that case, among the corresponding closed twin equivalence classes, we introduce the concepts of plain, compound and critical classes. The study of critical classes, together with properties of the Moore closure operator, allow us to correct a mistake in the proof of [P. J. Cameron, The power graph of a finite group, II, J. Group Theory 13 (2010), 6, 779–783; Theorem 2] and to deduce a simple algorithm to reconstruct the directed power graph of a finite group from its undirected counterpart, as asked in [P. J. Cameron, Graphs defined on groups, Int. J. Group Theory 11 (2022), 2, 53–107; Question 2].

The aim of this paper is to confirm a conjecture of Isaacs and Navarro from 1995, which asserts that, for any π ′ \pi^{\prime} -subgroup 𝑄 of a 𝜋-separable group 𝐺, the number of π ′ \pi^{\prime} -weights of 𝐺 with 𝑄 as the first component is larger than or equal to the number of irreducible 𝜋-partial characters of 𝐺 with 𝑄 as their vertex. We also give a sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the 𝜋-version of the Alperin weight conjecture.