Simple group

In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself.

Related Topics:
Mathematics - Group - Trivial group - Normal subgroup

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Despite the name, simple groups are far from "simple". The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan-Hölder theorem. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982.

Related Topics:
Finite simple groups - Prime number - Integer - Jordan-Hölder theorem - Classification of finite simple groups

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The only abelian simple groups are the cyclic groups of prime order. Apart from those (for which the term "simple" is appropriate), the smallest simple groups are three of order 60 (including the alternating group) and two of order 168 (including PSL(2,7)).

Related Topics:
Abelian - Cyclic group - Prime - Order - PSL(2,7)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore every finite simple group has even order unless it is cyclic of prime order.

Related Topics:
Theorem - Feit - Thompson - Solvable

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these.

Related Topics:
Simple Lie group - Thompson groups

~ ~ ~ ~ ~ ~ ~ ~ ~ ~