A simple group is basically a group having no proper nontrivial normal subgroups. For example, A_{5} 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 {e_{G}} and G are the only normal subgroups of G.

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

## Examples of Simple Groups

- The alternating group A
_{n}for n≥5 is a simple group. Note that A_{5}is the example of the smallest non-abelian simple group of order 60. - Z/pZ is a simple group where p is a prime number.
**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, ExamplesCyclic Group: The definition, properties, and related theorems on cyclic groups are discussed.Left Coset and Right Coset: Definition, Examples, Properties and TheoremsNormal SubgroupsFirst Isomorphism Theorems of GroupsKernel 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 Z_{p} for some prime p.

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

- |G| is divisible by at least three distinct primes (proved by Burnside, 1904).
- |G| is even (proved by Feit-Thompson, 1963).
- 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.
- A
_{n}, 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.