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