Related papers: The exponent of the non-abelian tensor square and …

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that if $G$ is an odd order finite non-abelian monolithic $p$-group such…

Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic…

In the paper we obtain some new upper bounds for exponential sums over multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1}, p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an application we prove that…

A result of the author shows that the behavior of Gowers norms on bounded exponent abelian groups is connected to finite nilspaces. Motivated by this, we investigate the structure of finite nilspaces. As an application we prove inverse…

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H O_{\pi}(G)/ O_{\pi}(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is…

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group $G$. Suppose that $G$ is a compact pro-$p$ $p$-adic Lie group with no torsion and that it contains a closed normal subgroup $H$ such…

A non-abelian generalisation of a theory of gravity coupled to a 2-form gauge field and a dilaton is found, in which the metric and 3-form field strength are Lie algebra-valued. In the abelian limit, the curvature with torsion is self-dual…

Let $G$ be a nonabelian group, $A\subseteq G$ an abelian subgroup and $n\geqslant 2$ an integer. We say that $G$ has an $n$-abelian partition with respect to $A$, if there exists a partition of $G$ into $A$ and $n$ disjoint commuting…

We prove that if $G$ is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of $G$ in the following four cases: (i) $G$ is metabelian, (ii) $\exp G=4$, (iii) $G$ is nilpotent of class $\le 5$, or (iv) $G$ is a…

The present paper is a note on the tensor degree of finite groups, introduced recently in literature. This numerical invariant generalizes the commutativity degree through the notion of nonabelian tensor square. We show two inequalities,…

Let $G$ be a finite group and $\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$, where $o(a)$ denotes the order of $a$ in $G$ and $\exp(G)$ denotes the exponent of $G$. Under a natural hypothesis, in this note we determine the groups $G$ such…

The Erd\H{o}s-Ginzburg-Ziv constant of an abelian group $G$, denoted $\mathfrak{s}(G)$, is the smallest $k\in\mathbb{N}$ such that any sequence of elements of $G$ of length $k$ contains a zero-sum subsequence of length $\exp(G)$. In this…

Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of…

Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial…

The genus spectrum of a finite group $G$ is the set of all $g$ such that $G$ acts faithfully on a compact Riemann surface of genus $g$. It is an open problem to find a general description of the genus spectrum of the groups in interesting…

In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a…

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all…

We consider on Riemannian manifolds the nonlinear evolution equation \begin{equation*} \partial _{t}u=\Delta _{p}(u^{1/(p-1)}), \end{equation*}% where $p>1$. This equation is also known as a doubly non-linear parabolic equation or…

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower…

Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on…