Can someone help me with proof of the next statement?
Suppose that the group $G$ acts transitively on the set $A$, and let $H$ be the stabiliser of $a\in A$.Then $G$ acts primitively on $A$ if and only if $H$ is a maximal subgroup of $G$.
Proof - Assume that $H$ is not maximal, and choose a subgroup $K$ with $H < K < G$. Then the points $A$ are in bijection with the (right) cosets of $H$ in $G$. Now the cosets of $K$ in G are unions of $H$-cosets, so correspond to sets of points, each set containing $|K:H|$ points.
This part is problem:
But the action of $G$ preserves the set of $K$-cosets, so the corresponding sets of points form a system of imprivitivity for G on $A$.
I don't understand the last part. Can someone help?
Thanks!