Block system

Remarks

Each part of a block system is a block. Conversely, if $G$ is transitive and $B$ is a block, then $\set{gB : g \in G}$ is a block system. For transitive $G$ all parts of a block system have the same size, so for finite $\Omega$ this size divides $\abs{\Omega}$. The partition into singletons and the partition $\set{\Omega}$ are the trivial block systems. A transitive action whose only block systems are trivial is primitive.

Example: the eight symmetries of a square acting on the corners $1, 2, 3, 4$ in cyclic order, and $\mathcal{P} = \set{\set{1, 3}, \set{2, 4}}$. Each symmetry maps a pair of opposite corners to a pair of opposite corners, so it maps each part of $\mathcal{P}$ to a part of $\mathcal{P}$.

Non-example: for the same action, $\mathcal{P}' = \set{\set{1, 2}, \set{3, 4}}$. The quarter turn $(1\,2\,3\,4)$ maps $\set{1, 2}$ to $\set{2, 3}$, which is not a part of $\mathcal{P}'$.

If $G$ is transitive and $\omega \in \Omega$, the blocks containing $\omega$ correspond to the subgroups $H$ with $G_\omega \le H \le G$, where $G_\omega$ is the stabiliser: $H$ gives the block $H\omega$. The orbits of a normal subgroup of $G$ form a block system.