Related papers: Antiassociative Groupoids

(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a…

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k + \ldots + \alpha_1 x + \alpha_0,$$ where $\alpha_0, \ldots , \alpha_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the…

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$…

We prove that the set of all orders of finite algebras in the groupoid variety of anti-rectangular Abel Grassmann bands consists of all powers of four. We also prove that any groupoid anti-isomorphic to a finite or countable…

Let G be a group and let k be a cardinal. A subset A of G is called left (right) k-large if there exists a subset F of G such that |F| < { and G = FA (G = AF). We say that A is k-large if A is left and right k-large. It is known that every…

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then…

An ample groupoid is said to be AF if it is a directed union of compact open principal subgroupoids. In this paper, we provide a complete homological characterization of these groupoids. Specifically, we prove that an ample groupoid is AF…

In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X, G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular,…

In this paper, we show that every $(3k-3)$-edge-connected graph $G$, under a certain condition on whose degrees, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$, $|d_{G_i}(v)-d_G(v)/k|< 1$,…

In a 2018 paper, Cameron and Semeraro posed the problem of finding all group-graph reciprocal pairs. In this paper, we make a significant contribution to finding all such pairs. A group and graph form a reciprocal pair if they satisfy the…

Let $G$ be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to $3$. We construct a canonical correspondence between irreducible characters of degree coprime to $3$ of $G$ and those…

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$.…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

Let $c:\mathcal{G}\to\R$ be a cocycle on a locally compact Hausdorff groupoid $\mathcal{G}$ with Haar system. Under some mild conditions (satisfied by all integer valued cocycles on \'{e}tale groupoids), $c$ gives rise to an unbounded odd…

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified…

We show that, if $w_1, \ldots , w_6$ are words which are not an identity of any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$ for all (non-abelian) finite simple groups $G$. In particular, for every word $w$,…

Let $G$ be a finite group acting faithfully on a finite set $\Omega$. For a positive integer $k$, $G$ acts naturally on the Catesian product $\Omega^k := \Omega \times ...\times \Omega$. In this paper, we prove that finite nilpotent group…

A finite group $G$ is called $k$-factorizable if for any factorization $|G|=a_1\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $|A_i|=a_i$ such that $G=A_1\cdots A_k$. We say that $G$ is \textit{multifold-factorizable} if $G$…

Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved…

We show that for non-conjugate subgroups $G_1$ and $G_2$ of a finite group $G$ there exists an extension of $G$ (by a finite group) in which the pre-images of $G_1$ and $G_2$ are not isomorphic. This allows us to show that $\mathbb Z$-coset…