# Simple non-abelian group

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

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*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 simple non-abelian group if:

- It is simple, i.e., it has no proper nontrivial normal subgroups
- It is not abelian, i.e., it is
*not*true that any two elements in the group commute.

## Examples

The smallest simple non-abelian group is alternating group:A5. See A5 is the simple non-abelian group of smallest order. See also:

There are also infinite simple non-abelian groups (in fact, any infinite simple group *must* be non-abelian). Examples are the finitary alternating groups on infinite sets (see finitary alternating groups are simple) and the projective special linear groups on infinite fields (see projective special linear groups are simple).

## Facts

### Every subgroup-defining function gives trivial group or whole group

Since any subgroup-defining function (such as the center, the derived subgroup, the Frattini subgroup, the Fitting subgroup etc.) returns a characteristic subgroup of the whole group, and since every characteristic subgroup is normal, any subgroup obtained via a subgroup-defining function must be either trivial or the whole group. This, combined with the fact that the group is non-Abelian, tells us the following:

- The center must be trivial -- in other words, the group is centerless
- The derived subgroup must be the whole group -- in other words, the group is perfect

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

infinite simple group | infinite and a simple group | the only simple abelian groups are the groups of prime order, which are all finite. | there are finite simple non-abelian groups | |

finite simple non-abelian group | finite, simple, and non-abelian | direct | there are infinite simple groups, which are hence non-abelian. |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

perfect group | equals its own derived subgroup | simple and non-abelian implies perfect | SL(2,5) is an example of a perfect group that is not simple. | Quasisimple group, Semisimple group|FULL LIST, MORE INFO |

centerless group | its center is a trivial group | simple and non-abelian implies centerless | symmetric group:S3 is a centerless group that is not simple. | |FULL LIST, MORE INFO |

quasisimple group | perfect group whose inner automorphism group is simple non-abelian. | simple and non-abelian implies quasisimple | SL(2,5), and more generally, any universal central extension of a simple non-abelian group that is not Schur-trivial, gives an example. | |FULL LIST, MORE INFO |

semisimple group | central product of quasisimple groups. | (via quasisimple) | (via quasisimple) | |FULL LIST, MORE INFO |

almost simple group | can be embedded between a simple non-abelian group and its automorphism group | (by definition) | symmetric group:S5 is an almost simple group that is not simple. More generally, the automorphism group of any simple non-abelian group that is not complete. | |FULL LIST, MORE INFO |

capable group | can be expressed as the inner automorphism group of some group. | (via centerless) | (via centerless) | Centerless group|FULL LIST, MORE INFO |