# Simple non-abelian group

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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.

## Facts

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

Since any subgroup-defining function (such as the center, the commutator 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 commutator 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 |