For a finite group $G$ denote by $N\left(G\right)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N\left(G\right)=N\left(L\right)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z\left(G\right)=1$ and $N\left(G\right)=N\left(A_i\right)$ is necessarily isomorphic to $A_i$, where $i\in \{2p,2p+1\}$.

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $G$ be a finite group and let $N\left(G\right)$ denote the set of conjugacy class sizes of $G$. Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N\left(G\right)=N\left(S\right)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong \mathrm{Sym}(p+1)$ if and only if $\left|G\right|=(p+1)!$ and $G$ has a special conjugacy class of size $(p+1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N\left(G\right)=N\left(\mathrm{Sym}\right(p+1\left)\right)$, then...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...