$p$-solvability of regular equations over unitriangular groups over prime finite fields

TL;DR

This paper proves that certain regular equations over unitriangular groups over finite fields are solvable in explicitly constructible overgroups, extending known results and providing explicit solutions.

Contribution

It establishes solvability of regular equations of specific exponents over UT$_n(\

Findings

Regular equations of exponent $rp^s$ are solvable over UT$_n(\

concrete results include solutions in overgroups UT$_{(n-1)p^s + 1}(\\mathbb{F}_p)$ and UT$_{2p^s + 1}(\\mathbb{F}_p)$.

The proofs are constructive, enabling explicit solutions.

Abstract

An equation over a group with one unknown is called regular if the exponent sum of the unknown is nonzero. In this paper we prove that some regular equations of exponent $rp^s$, where $r \in \mathbb{Z}$, $s \in \mathbb{N}$, $\gcd(r,p)=1$, over the group UT$_n(\mathbb{F}_p)$ ($n \geq 2$) are solvable in an overgroup isomorphic to UT$_{(n-1)p^s + 1}(\mathbb{F}_p)$. Applying this for $n=3$ we prove that any regular equation of exponent $rp^s$ over the Heisenberg $p$-group UT$_3(\mathbb{F}_p)$ is solvable in an overgroup isomorphic to UT$_{2p^s + 1}(\mathbb{F}_p)$. The proofs of these results are constructive and allow to obtain solutions of equations in explicit form.