Related papers: Frattini-injectivity and Maximal pro-$p$ Galois gr…

In the first half of this paper, we outline the construction of a new class of abelian pro-$p$ groups, which covers all countably-based pro-$p$ groups. In the second half, we study them, and classify them up to topological isomorphism and…

We say that a group $G$ is of \textit{profinite type} if it can be realized as a Galois group of some field extension. Using Krull's theory, this is equivalent to the ability of $G$ to be equipped with a profinite topology. We also say that…

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…

Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the…

In this article we trace the genesis of a theorem that gives for the first time examples of Galois group $G_S$ of the maximal $p$-extension of $\mathbb{Q}$, unramified outside a finite set of primes not containing $p$, that are of…

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends…

We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific…

Let $p$ be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to the strengthened version of Mina\v{c}--T\^an's Massey Vanishing Conjecture in the case of finitely generated maximal…

A pro-p Cappitt group is a pro-p group G such that the subgroup topologically generated by all non-normal closed subgroups is a proper subgroup of G. In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is…

We prove that the geometric etale fundamental group of a (geometrically connected) rigid smooth $p$-adic affinoid curve is a semi-direct factor of a certain profinite free group. We also prove that the maximal pro-$p$ (resp. maximal…

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is…

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely…

We study the class of finite groups $G$ satisfying $\Phi (G/N)= \Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe…

Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of $G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is not abelian, then $G$ is called an $\mathcal{A}_t$-group. In this paper, some information…

Let $p$ be an odd prime number and $\zeta_{p} := \exp(2\pi i/p)$. Then, it is well-known that the $A_{p-1}$-root lattice can be realized as the (Hermitian) trace form of the $p$-th cyclotomic extension $\mathbb{Q}(\zeta_{p})/\mathbb{Q}$…

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these…

A group is called metahamiltonian if all non-abelian subgroups of it are normal. This concept is a natural generation of Hamiltonian groups. In this paper, a complete classification of finite metahamiltonian $p$-groups is given.

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions.…

A finite separable extension $E$ of a field $F$ is called primitive if there are no intermediate extensions. It is called solvable if the group $\mathrm{Gal}(\hat E|F)$ of automorphisms of its galoisian closure $\hat E$ over $F$ is…