# 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.

## 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.
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