A subgroup $H$ of a finite group $G$ is said to be $ss$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $s$-permutable in $K$. In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $ss$-supplemented in $G$, and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $ss$-supplemented in $G$. These results...

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...

A theorem of Burnside asserts that a finite group $G$ is $p$-nilpotent if for some prime $p$ a Sylow $p$-subgroup of $G$ lies in the center of its normalizer. In this paper, let $G$ be a finite group and $p$ the smallest prime divisor of $\left|G\right|$, the order of $G$. Let $P\in {\mathrm{Syl}}_p\left(G\right)$. As a generalization of Burnside’s theorem, it is shown that if every non-cyclic $p$-subgroup of $G$ is self-normalizing or normal in $G$ then $G$ is solvable. In particular, if $P\ncong \langle a,b|a^{p^{n-1}}=1,b^2=1,b^{-1}ab=a^{1+p^{n-2}}\rangle$, where $n\ge 3$ for $p>2$ and $n\ge 4$ for $p=2$, then $G$ is $p$-nilpotent or $p$-closed. ...

Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D\left(G\right)$) the crossed product of $C\left(G\right)$ and $ℂH$ (or $ℂG$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D\left(G\right),e\rangle$ generated by $D\left(G\right)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D\left(G\right)$ for a certain conditional expectation $E$ of $D\left(G\right)$ onto $D(G;H)$. Let us call $\langle D\left(G\right),e\rangle$ the basic construction from the conditional expectation $E:D\left(G\right)\to D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)⋊ℂG$, and proves that there is an algebra isomorphism between...

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...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................