The main purpose of this article is to study pro-p groups with quadratic Fp-cohomology algebra, i.e. H • -quadratic pro-p groups. Prime examples of such groups are the maximal Galois pro-p groups of fields containing a primitive root of unity of order p.We show that the amalgamated free product and HNN-extension of H • -quadratic pro-p groups is H • -quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-p groups that yields many new examples of H • -quadratic groups: p-RAAGs. These examples generalise right angled Artin groups in the category of pro-p groups. Finally, we explore "Tits alternative behaviour" of H • -quadratic pro-p groups.
We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.
In this paper, we address the following question: when is a finite p-group G self-similar, i.e. when can G be faithfully represented as a self-similar group of automorphisms of the p-adic tree? We show that, if G is a self-similar finite p-group of rank r, then its order is bounded by a function of p and r. This applies in particular to finite p-groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass 1, we can fully answer the question above: a p-group of maximal class G is self-similar if and only if it contains an elementary abelian maximal subgroup over which G splits. Furthermore, in that case the order of G is at most p p+1 , and this bound is sharp.2010 Mathematics Subject Classification. Primary 20E08.
Abstract. We study infinitely iterated wreath products of finite permutation groups w.r.t. product actions. In particular, we prove that, for every non-empty class of finite simple groups X , there exists a finitely generated hereditarily just infinite profinite group W with composition factors in X such that any countably based profinite group with composition factors in X can be embedded into W . Additionally we investigate when infinitely iterated wreath products of finite simple groups w.r.t. product actions are co-Hopfian or non-co-Hopfian.
Let S be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in S is topologically finitely generated, provided that the actions of the groups in S are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in S are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of special sequences S. Notation 1. All the actions will be right actions and e will stand for the identity element of a group. We will write n = {1, . . . , n}.Notation 2. Let d be an integer. In all of this paper we will denote by S a sequence of finite transitive permutation groups {S k ≤ Sym(m k )} k∈N such that each S k is perfect and at most d-generated as an abstract group.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.