Primitive group action

This article defines a group action property or a property of group actions: a property that can be evaluated for a group acting on a set.
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Definition

Symbol-free definition

A transitive group action on a set with more than one element is termed primitive if it satisfies the following equivalent conditions:

The isotropy subgroup of any point is a maximal subgroup
There is no proper nontrivial block for the group. Here, a block for a group action on a set is a subset of the set that is disjoint from any other set in its orbit.

Definition with symbols

Let $G$ be a group acting transitively on a set $S$ that has more than one element. Then, the action of $G$ on $S$ is termed primitive if the following equivalent conditions are satisfied:

For any $s\in S$ , the group $Stab_{G}(s)$ is a maximal subgroup of $G$ .
The only blocks for $S$ are the improper block (where the whole $S$ is the block) and the trivial block (which comprises only one element).

Equivalence of definitions

The equivalence of the two forms of the definition (the one in terms of the isotropy subgroup and the onein terms of blocks) arises from the following basic fact:

There is a one-to-one correspondence between block decompositions of the set under the group action, and subgroups containing the isotropy subgroup

Relation with other properties

Stronger properties

Doubly transitive group action

Weaker properties

Transitive group action

Resultant group properties

A primitive group is a group which possesses a faithful primitive group action. If we view a primitive group along with its group action, we are essentially viewing it as a subgroup of the symmetric group.

Subgroups of the symmetric group

We are usually interested in the theory of faithful primitive group actions, or equivalently, subgroups of the symmetric group whose natural action is primitive.

In particular we have results on: