# Finite simple group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and simple group

View other group property conjunctions OR view all group properties

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Finite simple group, all facts related to Finite simple group) |Survey articles about this | Survey articles about definitions built on this

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

View a complete list of semi-basic definitions on this wiki

*This article is about a term related to the Classification of finite simple groups*

## Definition

### Symbol-free definition

A group is said to be a **finite simple group** if it satisfies both these conditions:

- It is a finite group, viz its order (the cardinality of its underlying set) is a finite integer
- It is a simple group, viz it has no proper nontrivial normal subgroup

## Relation with other properties

### Stronger properties

### Weaker properties

## Related notions

### Classification problem

The Classification problem is the problem of obtaining a complete description or classification of all the finite simple groups. The Classification Problem has now been solved, viz it has been shown that every finite simple group is one of these:

- A cyclic group of prime order
- An alternating group
- A classical group viz a group obtained as a subquotient of one of the typical linear groups (special linear group, orthogonal group, unitary group, symplectic group)
- An exceptional group of Lie type
- A sporadic simple group

`Further information: Classification of finite simple groups`

### Feasibility

A natural number is said to be simple-feasible if it occurs as the order of a simple group. One of the important initial questions in the classification of simple groups is: what are the simple-feasible numbers?

To answer this question, we need a mix of techniques from Sylow theory, representation theory, group actions, and of course basic arguments with subgroups.

`Further information: simple-feasible number`