G-空间中的万花筒
Kaleidoscopical Configurations in G-spaces
摘要：
设$G$是一个群，$X$是一个$G$ -空间。如果存在满射着色$χ:X\to Y$使得$χ$对每个子集$gF$，$g\in G$的限制是双射，则称$X$的子集$F$为万花筒构形.给出了任意$G$ -空间中万花筒构形的构造，发展了阿贝尔群（被认为是具有作用$(g,x)\mapsto g+x$的$G$ -空间）的万花筒技巧，并描述了$N=p^m$或$N=p_1... p_k$阶循环群中的万花筒构形，其中$p$是素数，$p_1,...,p_k$是不同素数.设$G$是一个群，$X$是一个$G$ -空间。如果存在着色$χ:X\rightarrow C$使得$χ$对每个子集$gF$的限制$g\in G$是双射，则$X$的子集$F$称为万花筒配置。给出了任意$G$ -空间中万花筒构形的一种构造（称为分裂构造），将有限阿贝尔群$G$中万花筒构形的刻画问题归结为将$G$分解为两个子集的问题，并描述了具有有限距离尺度的等距齐性超度量空间中的所有万花筒构形.并在欧氏空间$R^n$中构造了基数连续统的$2^c$（不可分裂）万花筒构型.
英文摘要：
Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a surjective coloring $χ:X\to Y$ such that the restriction of $χ$ on each subset $gF$, $g\in G$ is a bijection. We give some constructions of kaleidoscopical configurations in an arbitrary $G$-space, develop some kaleidoscopical technique for Abelian groups (considered as $G$-spaces with the action $(g,x)\mapsto g+x$), and describe kaleidoscopical configurations in the cyclic groups of order $N=p^m$ or $N=p_1... p_k$ where $p$ is prime and $p_1,...,p_k$ are distinct primes. Let $G$ be a group and $X$ be a $G$-space. A subset $F$ of $X$ is called a kaleidoscopical configuration if there exists a coloring $χ:X\rightarrow C$ such that the restriction of $χ$ on each subset $gF$, $g\in G$, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical configurations in an arbitrary $G$-space, reduce the problem of characterization of kaleidoscopical configurations in a finite Abelian group $G$ to a factorization of $G$ into two subsets, and describe all kaleidoscopical configurations in isometrically homogeneous ultrametric spaces with finite distance scale. Also we construct $2^c$ (unsplittable) kaleidoscopical configurations of cardinality continuum in the Euclidean space $R^n$.
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