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

Abstract

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.

12 pages

Cited by in corpus (1)

• On the density of images of the power maps in Lie groups