Related papers: The pro-$p$ group of upper unitriangular matrices

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$,…

Recently the first example of a family of pro-$p$ groups, for $p$ a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with…

For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one…

We study asymptotic behavior of the dimensions of the homology groups of subgroups of finite index in finitely generated subgroups of pro-$p$ extension of centralizers of free pro-$p$ groups. We also prove group theoretic structure…

Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le \Phi(P)$…

The main purpose of this article is to study pro-$p$ groups with quadratic $\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups. Prime examples of such groups are the maximal Galois pro-$p$ groups of fields…

Let G be a noncompact connected Lie group and $\rho$ be the right Haar measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields which satisfy H\"ormander's condition, and let $\Delta=-\sum_{i=1}^qX_i^2$ be the…

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length of G is defined as the minimal…

We determine the Grothendieck ring of finite-dimensional comodules for the free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf algebra with bijective antipode and other related universal quantum groups. The results turn…

Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-$\mathbb{Z}_p$ pro-$p$ subgroup. We show that the completed group algebra of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a…

Firstly, we completely determine the self-similar Hausdorff spectrum of the group of $q$-adic automorphisms where $q$ is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p…

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index…

A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We…

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields.…

We describe the Sylow subgroups of Gal(Q) for an odd prime p, by observing and studying their decomposition as a semidirect product of Z_p acting on F, where F is a free pro-p group, and Z_p are the p-adic integers. We determine the finite…

We study 3-dimensional Poincar\'e duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index,…