Normal Series

Relevant Courses: Math 493. Resources Used: https://youtu.be/wE1kxAChirg?si=jqv2ogzeMJQiZOw6

Definition: Ascending Chain of Subgroups

Let $G$ be an arbitrary group. An ascending chain of subgroups is a chain of subgroups as follows: $(e)=G_0<G_1<\cdots <G_k=G,$ where each subgroup is a strict subgroup of the previous one. This is also called a subgroup series.

This then leads to the following definition of a Normal Series:

Definition: Normal Series

Let $G$ be an arbitrary group. A normal series of $G$ is an ascending chain of subgroups $(e)=G_0◃G_1◃\cdots ◃G_k=G,$

Where each group $G_i$ is a normal subgroup of $G_{i+1}$.

A normal series is also known as a subnormal series.