相关论文: On Graham's rearrangement conjecture over $\mathbb…

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this…

In Section 6 of the paper ``Fra\"{\i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic…

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the…

Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement…

We say that a finite almost simple $G$ with socle $S$ is admissible (with respect to the spectrum) if $G$ and $S$ have the same sets of orders of elements. Let $L$ be a finite simple linear or unitary group of dimension at least three over…

In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta(G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This answers, in a strong…

A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an ordering $(y_1, \ldots, y_k)$ of its elements such that the partial sums~$s_0, s_1, \ldots, s_k$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i y_j$ for $1 \leq i \leq…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…

A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is…

Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and…

Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r…

For a group $G$, $\mathcal{F}_G$ denotes the set of all non-empty finite subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times G$ to $\mathcal{F}_G\times \mathcal{F}_G$ and say that a macro-uniform mapping $f:…

In this paper we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$…

Let $G$ be a group and $S\subseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then {\it the Cayley graph ${\rm Cay}(G,S)$} is an undirected graph $\Gamma$ with the vertex set $V(\Gamma)=G$ and the edge set…

Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of $A$ is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime…

An orthomorphism of a finite group $G$ is a bijection $\phi\colon G\to G$ such that $g\mapsto g^{-1}\phi(g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $G$ is abelian, for any $k\geq 2$ dividing…

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in…

Let $\mathcal{CT}_n$ be the semigroup of full contraction mappings on $[n]=\{1,2,\ldots,n\}$, and let $\mathcal{OCT}_n$ and $\mathcal{ODCT}_n$ be the subsemigroups consisting of all order-preserving full contraction and subsemigroup of…

In this article, we first generalize Kaplansky's zero-divisor conjecture of group-rings $K[G]$ (with $K$ a field) to the more general setting of $G$-graded rings $R=\bigoplus\limits_{n\in G}R_{n}$ with $G$ a torsion-free group. Then we…

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If…