# Primitive group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

## Contents

## Definition

### Symbol-free definition

A group is said to be **primitive** if the following equivalent conditions hold:

- It has a core-free maximal subgroup
- It possesses a nontrivial faithful primitive group action

## Relation with other properties

### Stronger properties

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

finite simple group | finite and simple: nontrivial with no proper nontrivial normal subgroup | Simple group|FULL LIST, MORE INFO | ||

finite characteristically simple group | finite and characteristically simple: nontrivial with no property nontrivial characteristic subgroup | |FULL LIST, MORE INFO | ||

2-transitive group | has a 2-transitive action on a set | |FULL LIST, MORE INFO |

### Weaker properties

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

quasiprimitive group | possesses a faithful action whose restriction to every nontrivial normal subgroup is transitive | |FULL LIST, MORE INFO | ||

innately transitive group | possesses a faithful group action with a transitive minimal normal subgroup | |FULL LIST, MORE INFO | ||

Frattini-free group | the Frattini subgroup (the intersection of all maximal subgroups) is trivial | |FULL LIST, MORE INFO |