Simple Group: Definition, Examples, Properties, Classification

A simple group is basically a group having no proper nontrivial normal subgroups. For example, A5 is a simple group. In this post, we will learn about simple groups with examples, properties, and classification.

Definition of Simple Group

A group is called a simple group if its only normal subgroups are the trivial subgroup and the group itself.

In other words, a group G is called a simple group if {eG} and G are the only normal subgroups of G.

For example, Zp which is a cyclic group of order p is a simple group as it has no proper nontrivial normal subgroups.

Examples of Simple Groups

  1. The alternating group An for n≥5 is a simple group. Note that A5 is the example of the smallest non-abelian simple group of order 60.
  2. Z/pZ is a simple group where p is a prime number.
  3. Infinite simple group: The infinite alternating group A is a simple group. A is the group of even finitely supported permutations of the integers.
Also Read:

Order of a Group: The order of a group and of its elements are discussed here with formulas.

Abelian Group: Definition, Properties, Examples

Cyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.

Left Coset and Right Coset: Definition, Examples, Properties and Theorems

Normal Subgroups

First Isomorphism Theorems of Groups

Kernel of a Group Homomorphism

Properties of Simple Groups

Simple groups have many properties; for example, finite abelian or non-abelian simple groups can be characterized. Below are a few properties of simple groups.

A simple group has only two normal subgroups; they are {1} and the group itself.

If G is a non-trivial simple group with normal subgroup H, then we must have that H is trivial or H=G.

If G is a finite abelian simple group, then it is isomorphic to Zp for some prime p.

If G is a finite non-abelian simple group, then we have

  1. |G| is divisible by at least three distinct primes (proved by Burnside, 1904).
  2. |G| is even (proved by Feit-Thompson, 1963).
  3. Let p be a prime dividing |G|. If there are no divisors (≠ 1) of |G| which are ≡ 1 (mod p), then G is not simple. (proved by Sylow, 1872).

Classification of Simple Groups

The major classification of finite simple groups was done in the 1970s and early ’80s and it was completely classified in 2008.

A finite simple group is isomorphic to one of the groups listed below.

  • Z/pZ, a cyclic group of prime order p.
  • An, the alternating group with n≥5.
  • a simple group of “Lie type”.
  • one of the 26 “sporadic” groups that are not of the above types.