Related papers: Finiteness of profinite groups with a rational pro…

A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is…

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p…

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an…

The number of maximal abelian subgroups of a finite p-group is shown to be congruent to 1 modulo p.

We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups…

We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on…

In this note, we study the finite groups with the number of cylic subgroups no greater than 6.

We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the \'etale…

Let $\mathfrak C$ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group $G$ and an element $x\in G$, we denote by $P_{\mathfrak{C}}(x,G)$ the probability that $x$ and a randomly…

We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the…

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that…

We prove that the class of closed subgroups of free profinite monoids is precisely the class of projective profinite groups. In particular, the profinite groups associated to minimal symbolic dynamical systems by Almeida are projective. Our…

We prove that the subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group are finite sums of Euler products of cone integrals over $\mathbb{Q}$ and we deduce from this that they have rational…

We prove that finitely generated free metabelian groups $\Psi_n$ are profinitely rigid in the absolute sense: they are distinguished by their finite quotients among all finitely generated residually finite groups. The proof is based on a…

The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. We show that every non-positive integer is the…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

We describe finite soluble groups in which every $n$-maximal subgroup is $\mathfrak F$-subnormal.

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…