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

Abstract

Abstract Let G be a group containing a nilpotent normal subgroup N with central series { N j } {\{N_{j}\}} such that each N j / N j + 1 {N_{j}/N_{j+1}} is an 𝔽 {\mathbb{F}} -vector space over a field 𝔽 {\mathbb{F}} and the action of G on N j / N j + 1 {N_{j}/N_{j+1}} induced by the conjugation action is 𝔽 {\mathbb{F}} -linear. For k ∈ ℕ {k\in\mathbb{N}} we describe a necessary and sufficient condition for all elements from any coset xN, x ∈ G {x\in G} , to admit kth roots in G, in terms of the action of x on the quotients N j / N j + 1 {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 𝔽 {\mathbb{F}} -algebraic groups we also characterise the property in terms of centralisers of elements. For a class of Lie groups, it is shown that surjectivity of the kth power map, k ∈ ℕ {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.