G-invariant block systems are exactly the invariant equivalence relations

For the forward direction, let $\mathcal{B}$ be a $G$-invariant partition into blocks. The relation ${\sim }_{\mathcal{B}}$ is reflexive because every point lies in its own part, symmetric because “lying in the same part” is symmetric, and transitive because two parts that meet are equal.

For the converse direction, let $\sim$ be a $G$-invariant equivalence relation. Its equivalence classes partition $\mathrm{\Omega }$: every point lies in its own class, and two classes that meet are equal because symmetry and transitivity identify every element of one with every element of the other.

For the converse direction, fix an equivalence class $C=[x]$. Invariance gives $g\cdot C=[g\cdot x]$ for every $g\in G$, so $G$ permutes the equivalence classes. In particular, if $(g\cdot C)\cap C\mathrm{\ne }\mathrm{\varnothing }$, the two equivalence classes are equal; otherwise they are disjoint. Thus every class is a block and the class partition is $G$-invariant.

For the forward direction, if $x{\sim }_{\mathcal{B}}y$ and both lie in a part $B$, then $g\cdot x$ and $g\cdot y$ lie in the part $g\cdot B$ of the same partition by its $G$-invariance. Thus ${\sim }_{\mathcal{B}}$ is $G$-invariant.

Step 1.3 shows that the equivalence classes form a $G$-invariant block system, and steps 1.1 and 2.1 give the converse construction.