Primitive group
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 properties
VIEW 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
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 |