Related papers: Polylogarithmic bounds in the nilpotent Freiman th…

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating…

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue…

In this paper, we prove that the intersection of all non-nilpotent maximal subgroups of a non-solvable group containing the normalizer of some Sylow subgroup is nilpotent, which provides an extension of Shlyk's theorem.

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

Let $K/k$ be a pro-$p$-extension over a number field $k$ whose Galois group is finitely generated and $k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$ an ascending sequence of intermediate fields of $K/k$ such that $k_n/k$ is…

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $\mathcal C\!\ell_{\mathbf K}$ of a number field ${\mathbf K}$ may be generated using prime ideals whose norm is bounded by $12\log^2\Delta_{\mathbf K}$, and by…

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and…

We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove the lower bound part of the conjecture for…

Let $G$ be a finite group and let $\mathscr{F}$ be a family of subgroups of $G$. We introduce a class of $G$-equivariant spectra that we call $\mathscr{F}$-nilpotent. This definition fits into the general theory of torsion, complete, and…

We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $ R_0 > \exp(\exp(Cd^C))$ for which the number of elements in a ball of radius $R_0$ in a…

We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper…

Let $G$ be a finite solvable group and $H$ a non-normal core-free subgroup of $G$. We show that if the normalizer of any non-trivial normal subgroup of $Fit(H)$ is equal $H$, then $H$ has a nilpotent normal complement $K$ such that $G=KH$…

We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by H\"ormander-Mikhlin criterion from classical harmonic analysis, although it…

Let $G =<S>$ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$…

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative…

When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal…

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete…

In 1974, Helmut Wielandt proved that in a finite group $G$, a subgroup $A$ is subnormal if and only if it is subnormal in every $\seq{A,g}$ for all $g\in G$. In this paper, we prove that the subnormality of an odd order nilpotent subgroup…

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely…

Let $G$ be a group and let $A\subseteq G$ be non-empty. We call $A$ an asymptotic $(r,l)$-approximate group if, for a fixed dilation factor $r$, the larger product sets $A^{hr}$ can, for all sufficiently large $h$, be covered by a bounded…