Class preserving automorphisms of unitriangular groups

TL;DR

This paper investigates automorphisms of certain quotients of unitriangular groups, showing that class-preserving automorphisms are inner over prime fields and explicitly calculating automorphism groups for specific cases.

Contribution

It characterizes class-preserving automorphisms of quotients of unitriangular groups and computes their automorphism groups in particular instances.

Findings

Class-preserving automorphisms are inner over prime fields.

Explicit automorphism groups are computed for specific quotients.

Automorphism structure depends on the field's nature and the group's parameters.

Abstract

Let $\textrm{UT}_n (K)$ be a unitriangular group over a field $K$ and $\Gamma_{n,k} := \textrm{UT}_n (K)/ \gamma_k(\textrm{UT}_n (K))$, where $\gamma_k (\mathrm{UT}_n(K))$ denotes the $k$-th term of the lower central series of $\mathrm{UT}_n (K)$, $2 \le k \le n$. We prove that the group of all class preserving automorphisms of $\Gamma_{n,k}$ is equal to $\Inn(\Gamma_{n,k})$ if and only if $K$ is a prime field. Let $G_n^{(m)} := \mathrm{UT}_n (\mathbb{F}_{p^m}) / \gamma_3 (\mathrm{UT}_n(\mathbb{F}_{p^m}))$. We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of $G_n^{(m)}$.