Let $G$ be a group acting transitively on a finite set $X$ with $|X|>1$. Fix $x \in X$, and define the stabiliser

\begin{align*} G_x := \{g \in G : g \cdot x = x\}. \end{align*}

A subset $B \subset X$ is called a block for the action if $B \neq \varnothing$ and, for every $g \in G$, either $g \cdot B = B$ or $(g \cdot B) \cap B = \varnothing$. The action is primitive if its only blocks are the singleton subsets of $X$ and $X$ itself. Then the action is primitive if and only if $G_x$ is a maximal proper subgroup of $G$.