Adjunction of roots to unitriangular groups over prime finite fields

TL;DR

This paper explores how to embed unitriangular groups over finite fields into larger such groups, enabling roots of elements and embedding wreath products, advancing understanding of their algebraic structure.

Contribution

It constructs explicit embeddings of UT$_n(F_p)$ into larger UT$_m(F_p)$ groups, allowing roots and wreath product embeddings, which was not previously established.

Findings

Embedded UT$_n(F_p)$ in UT$_m(F_p)$ with roots for all elements.

Constructed embedding of wreath product UT$_n(F_p) \, wr \, C_{p^s}$ into UT$_m(F_p)$.

Provided explicit formulas for embeddings based on parameters n, p, s.

Abstract

In this paper we study embeddings of unitriangular groups UT$_n(\mathbb{F}_p)$ arising under adjunction of roots. We construct embeddings of UT$_n(\mathbb{F}_p)$ in UT$_m(\mathbb{F}_p)$, for $n \geq 2$, $m=(n-1)p^s + 1$, $s \in \mathbb{Z}^+$, such that any element of UT$_n(\mathbb{F}_p)$ has a $p^s$-th root in UT$_m(\mathbb{F}_p)$. Also we construct an embedding of the wreath product UT$_n(\mathbb{F}_p) \wr C_{p^s}$ in UT$_m(\mathbb{F}_p)$, where $C_{p^s}$ is the cyclic group of order $p^s$.