# Innately transitive 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 primitivity|Find other variations of primitivity |

## Contents

## Definition

### 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.

### Definition with symbols

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