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

TL;DR

This paper characterizes when power maps are surjective in certain solvable groups with nilpotent normal subgroups, generalizing known results and applying to algebraic and Lie groups, with implications for exponentiality.

Contribution

It provides necessary and sufficient conditions for the surjectivity of power maps in a broad class of solvable groups, extending previous special case results.

Findings

Conditions for k-th roots in cosets of solvable groups.

Characterization of power map surjectivity in algebraic groups.

Implications for exponentiality of Lie 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.