Exercise 13, § 5

Let G be a transitive group of permutations of a set X and H the stabilizer of an element $x \in X$. Show the equivalence of the following properties:

(a) Every subgroup H' of G containing H is equal to H or G.

(b) Every subset Y of X such that, for all $g \in G$, $gY$ is either contained in Y or disjoint from Y, is equal to X or consists of a single element.

(If H' satisfies (a) and $Y = H'x$, then $gY = Y$ for all $g \in H'$ and $gY \cap Y = \varnothing$ for all $g \notin H'$. Conversely, if Y enjoys the property in (b), the set H' of $g \in G$ such that $gY = Y$ is a subgroup of G containing H.)

A transitive permutation group G satisfying (a) and (b) and not consisting only of the identity element is called primitive.