非交换性分裂
Splitting via Noncommutativity
摘要：
设$G$是一个非交换群，$n$是一个自然数。我们说$G$有严格的$n$ -分裂分解，如果它可以被划分为阿贝尔子群$A$和$n$非空子集$B_1, B_2, \ldots, B_n$的不交并，使得对于每个$i$和每个集合$B_i$，$|B_i| > 1$，没有两个不同的元素交换.我们证明了每个有限非交换群对于某些$n$都有严格的$n$ -分裂分解.我们对所有有限群$G$进行分类，直到同构，它们对$n = 1, 2, 3$有严格的$n$ -分裂分解.最后，我们证明了对于一个具有严格$n$ -分裂分解的非交换群$G$，其指数$|G:A|$被$n$的某个函数所限制.
英文摘要：
Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such that $|B_i| > 1$ for each $i$ and within each set $B_i$, no two distinct elements commute. We show that every finite nonabelian group has a strict $n$-split decomposition for some $n$. We classify all finite groups $G$, up to isomorphism, which have a strict $n$-split decomposition for $n = 1, 2, 3$. Finally, we show that for a nonabelian group $G$ having a strict $n$-split decomposition, the index $|G:A|$ is bounded by some function of $n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
