Given a finite group $G$, the smallest $n$ such that $G$ embeds into the symmetric group $S_n$ is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way.
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机译：给定一个有限的组$ G $，将使$ G $嵌入到对称组$ S_n $中的最小$ n $被称为最小度。大量的文献集中在最小程度与直接乘积之间的相互作用。本文将其扩展到涵盖大类的半直接产品。第1章提供了最低限度的背景知识-陈述和证明了许多基本定理，概述了相关的先前工作以及一些较小的原始结果。第2章计算了无限类半直接乘积的最小度数-特别是基本阿贝尔群的半直接乘积是按素数阶的组而不是基群的阶数进行划分的。这是使用向量空间理论建立的，其中包括许多新颖的技术。然后，通过以一种新的，可能具有普遍意义的方式回答最小程度领域中的一个现有问题来证明这项研究的实用性。
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