Roots of elements for groups over local fields

2025/03/21 by Parteek Kumar, Kumar, Parteek, Arunava Mandal +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #advanced mathematical theories #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2503.16987

Abstract

Let \mathbb F be a local field and G be a linear algebraic group defined over \mathbb F. For k∈\mathbb N, let g→ g^k be the k-th power map P_k on G(\mathbb F). The purpose of this article is two-fold. First, we study the power map on real algebraic group. We characterise the density of the images of the power map P_k on G(\mathbb R) in terms of Cartan subgroups. Next we consider the linear algebraic group G over non-Archimedean local field \mathbb F with any characteristic. If the residual characteristic of \mathbb F is p, and an element admits p^k-th root in G(\mathbb F) for each k, then we prove that some power of the element is unipotent. In particular, we prove that an element g∈ G(\mathbb F) admits roots of all orders if and only if g is contained in a one-parameter subgroup in G(\mathbb F). Also, we extend these results to all linear algebraic groups over global fields.

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