arXiv 1608.027012016-08-09math.GR

一类可解群的幂映射的满射性

On the surjectivity of the power maps of a class of solvable groups

S. G. Dani, Arunava Mandal

备注：: 12 pages

摘要：

设$G$是一个群，其中包含一个幂零正规子群$N$，其中心序列为$\{N_j\}$，使得每个$N_j/N_{j+1}$是域$\mathbb{F}$上的$\mathbb{F}$ -向量空间，并且共轭作用诱导的$G$对$N_j/N_{j+1}$的作用是$\mathbb{F}$ -线性的.对于$k\in \mathbb N$，我们描述了一个必要和充分条件的所有元素从任何陪集$xN$，$x\in G$，承认$k$ -th根在$G$，在行动方面的$x$上的同胚$N_j/N_{j+1}.$这产生特别是一个条件满射的权力地图，推广各种结果已知的特殊情况下。对于$\mathbb{F}$ -代数群，我们也用元素的中心化子证明了这一性质.对于一类李群，证明了$k$次幂映射$k\in \mathbb N$的满射性对于映射到群的可解根的限制是相同的.这些结果特别适用于李群指数性的研究。

英文摘要：

Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the conjugation action is $\mathbb{F}$-linear. For $k\in \mathbb N$ we describe a necessary and sufficient condition for all elements from any coset $xN$, $x\in G$, to admit $k$-th roots in $G$, in terms of the action of $x$ on the quotients $N_j/N_{j+1}.$ This yields in particular a condition for surjectivity of the power maps, generalising various results known in special cases. For $\mathbb{F}$-algebraic groups we also characterise the property in terms of centralizers of elements. For a class of Lie groups, it is shown that surjectivity of the $k$-th power map, $k\in \mathbb N$, implies the same for the restriction of the map to the solvable radical of the group. The results are applied in particular to the study of exponentiality of Lie groups.

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摘要：

英文摘要：

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