Simple Groups

Introduction

Simple groups are the "atoms" of group theory. They cannot be simplified further by forming non-trivial quotient groups and are fundamental to understanding the structure of all finite groups.

Definition

Definition 3.4: A group $G$ is simple if its only normal subgroups are the trivial subgroup ${e}$ and the group $G$ itself.

Key Examples

Cyclic Groups of Prime Order

The cyclic groups $Z_{p}$ for a prime $p$ are simple.

Proof: If $H$ is a non-trivial subgroup of $Z_{p}$ , then $| H |$ divides $p$ . Since $p$ is prime, $| H | = p$ , so $H = Z_{p}$ .

Alternating Groups

The alternating groups $A_{n}$ are simple for $n \geq 5$ .

Note: $A_{4}$ is not simple because it contains the Klein four-group as a normal subgroup.

Other Simple Groups

PSL groups: Projective special linear groups over finite fields
Sporadic groups: 26 exceptional simple groups
Lie type groups: Groups related to Lie algebras

Properties

No Non-Trivial Quotients

A group $G$ is simple if and only if it has no non-trivial quotient groups (other than $G$ itself).

No Non-Trivial Homomorphic Images

A group $G$ is simple if and only if every homomorphism from $G$ is either injective or trivial.

Composition Series

Simple groups are the building blocks in composition series. Every finite group can be "factored" into simple groups.

Classification of Finite Simple Groups

The classification of finite simple groups is one of the monumental achievements of 20th-century mathematics. It states that every finite simple group belongs to one of the following families:

1. Cyclic Groups of Prime Order

$Z_{p}$ where $p$ is prime.

2. Alternating Groups

$A_{n}$ for $n \geq 5$ .

3. Groups of Lie Type

Groups related to Lie algebras over finite fields, including:

PSL groups: Projective special linear groups
PSU groups: Projective special unitary groups
PSp groups: Projective symplectic groups
Orthogonal groups: Various orthogonal groups

4. Sporadic Groups

26 exceptional simple groups that don't fit into the other families, including:

Mathieu groups: $M_{11}, M_{12}, M_{22}, M_{23}, M_{24}$
Monster group: The largest sporadic simple group
Baby Monster: Second largest sporadic simple group

Applications

Application 1: Group Structure

Simple groups are essential for understanding the structure of all finite groups through composition series.

Application 2: Galois Theory

Simple groups are important in Galois theory, particularly in understanding the solvability of polynomial equations.

Application 3: Representation Theory

Simple groups are fundamental to representation theory and the study of group actions.

Application 4: Cryptography

Some cryptographic protocols are based on the difficulty of certain problems in simple groups.

Examples

Example 1: $Z_{5}$ is Simple

The group $Z_{5}$ has order 5, which is prime. Its only subgroups are ${0}$ and $Z_{5}$ itself, so it is simple.

Example 2: $A_{5}$ is Simple

The alternating group $A_{5}$ has order 60 and is simple. This is a key fact in the proof that the general quintic equation is not solvable by radicals.

Example 3: $S_{4}$ is Not Simple

The symmetric group $S_{4}$ is not simple because it contains the alternating group $A_{4}$ as a normal subgroup.

Example 4: $D_{4}$ is Not Simple

The dihedral group $D_{4}$ is not simple because it contains the subgroup ${e, r^{2}}$ as a normal subgroup.