Primitive group

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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:

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

Conjunction with other properties