Volume 27, Issue 5

The relational complexity of a subgroup 𝐺 of Sym ⁢ ( Ω ) \mathrm{Sym}({\Omega}) is a measure of the way in which the orbits of 𝐺 on Ω k \Omega^{k} for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between PSL n ⁢ ( F ) \mathrm{PSL}_{n}(\mathbb{F}) and PGL n ⁢ ( F ) \mathrm{PGL}_{n}(\mathbb{F}) , for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of F n \mathbb{F}^{n} . We also bound the relational complexity of all groups lying between PSL n ⁢ ( q ) \mathrm{PSL}_{n}(q) and P ⁢ Γ ⁢ L n ⁢ ( q ) \mathrm{P}\Gamma\mathrm{L}_{n}(q) , and generalise these results to the action on 𝑚-spaces for m ≥ 1 m\geq 1 .

Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function f : N → N f\colon\mathbb{N}\to\mathbb{N} such that, if 𝐺 is innately transitive of degree 𝑛 and the derangement graph of 𝐺 has no clique of size 𝑘, then n ≤ f ⁢ ( k ) n\leq f(k) . Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.

We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups M n M_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p < n p<n , we construct all irreducible representations of M n M_{n} by defining a standard form for the matrices of these representations.

Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ⁢ ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either { 1 , 5 , 5 3 } \{1,5,5^{3}\} or { 1 , 5 , … , 5 k } \{1,5,\ldots,5^{k}\} with k ≥ 1 k\geq 1 .

In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.

A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then G / F ⁢ ( G ) G/F(G) is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.

In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.

We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the Relative Primitivity Problem, using Shpilrain’s recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.

We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the Δ-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.

We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.