Doubly transitive implies primitive

Property-theoretic statement
For any group action, the property of being two-transitive is stronger than being primitive.

Verbal statement
Any doubly transitive group action is primitive.

Doubly transitive group action
A group acts on a set doubly transitively if, for every two pairs of distinct elements in the set, there is an element of the group mapping one pair of elements to the other.

Primitive group action
A group acts on a set primitively if the action cannot be decomposed into proper non-trivial blocks.

Proof
Let $$G$$ act on $$\Omega$$ doubly transitively. Take $$\alpha_1, \alpha_2 \in B$$ such that $$\alpha_1 \neq \alpha_2$$. Take $$\beta_1 = \alpha_1$$ and $$\beta_2 \in \Omega - B$$. Then $$\beta_1 \neq \beta_2$$. By double transitivity, $$\exists g \in G$$ such that $$g\alpha_1 = \beta_1$$ and $$g\beta_1 = g\beta_2$$. But then the setwise image $$gB$$ is not disjoint to $$B$$, so $$B$$ is not a block. Therefore, the action is primitive.