Complementation of Normal Subgroups

In Finite Groups

Joseph Kirtland

• Publisher: De Gruyter
• Year: 2017
• eBook ISBN: 9783110478921
• Edition: 1
• Topic: Mathematics
• Subtopic: Discrete Mathematics
• Index: Mathematics

Starting with the Schur-Zassenhaus theorem, this monograph documents a wide variety of results concerning complementation of normal subgroups in finite groups. The contents cover a wide range of material from reduction theorems and subgroups in the derived and lower nilpotent series to abelian normal subgroups and formations.

Contents
Prerequisites
The Schur-Zassenhaus theorem: A bit of history and motivation
Abelian and minimal normal subgroups
Reduction theorems
Subgroups in the chief series, derived series, and lower nilpotent series
Normal subgroups with abelian sylow subgroups
The formation generation
Groups with specific classes of subgroups complemented

1Prerequisites

It is assumed that the reader has a firm understanding of the fundamentals of finite group theory as would be found in a year-long introductory graduate course on the topic. However, some ideas are central to this work and are repeated here. In addition, a few topics required in subsequent chapters, but not typically found in a first course in group theory, are also presented. All groups are finite and the group operation will always be represented multiplicatively. Depending on the circumstances, the identity element of a group will represented by 1 or e.

1.1Products of subgroups

Definition 1.1. A group G is a product of two subgroups A and B if G = AB, where AB = {ab | a ∈ A and b ∈ B}.

When this occurs, each element g ∈ G can be written, although not necessarily uniquely, as g = ab with a ∈ A and b ∈ B. In general, given two subgroups A and B of a group G, the product AB need not be a subgroup of G. Clearly, AB is trivially a subgroup of G when G = AB. In cases...

Table of contents

• Cover
• Titlepage
• Copyright
• Dedication
• Preface
• Contents
• Notation
• 1 Prerequisites
• 2 The Schur-Zassenhaus theorem: A bit of history and motivation
• 3 Abelian and minimal normal subgroups
• 4 Reduction theorems
• 5 Subgroups in the chief series, derived series, and lower nilpotent series
• 6 Normal subgroups with abelian sylow subgroups
• 7 The formation generation
• 8 Groups with specific classes of subgroups complemented
• Bibliography
• Author index
• Subject index