Innately transitive group

Symbol-free definition
A group is said to be innately transitive if the following equivalent conditions hold:


 * It possesses a faithful group action and a minimal normal subgroup such that the restriction of the action to the minimal normal subgroup is a transitive group action. (In short, it possesses a faithful group action with a transitive minimal normal subgroup). Such a minimal normal subgroup is termed a plinth.
 * It can be expressed as the product of a minimal normal subgroup and a core-free subgroup.

The plinth theorem states that there are at most two minimal normal subgroups acting transitively, and that they must be isomorphic; in fact, they must be conjugate by a permutation on the set on which the group is acting.

Stronger properties

 * Weaker than::Simple group
 * Weaker than::Primitive group
 * Weaker than::Quasiprimitive group