Primitivity

The section is about decomposing group actions, assume $\Omega$ transitive.

Definition A non-empty set $\Gamma \subseteq \Omega$ is called a block if for all $g \in G$ either $\Gamma g = \Gamma$ or $\Gamma g \cap \Gamma = \{\}$. If $\Gamma$ is a block then the set $\Sigma = \Sigma(\Gamma) = \{\Gamma g \mid g \in G\}$ of all translates of $\Gamma$ is a block system.

Lemma A block system partitions $\Omega$. proof: Let $\alpha \in \Gamma$ and $\beta \in \Omega$ then $\beta = \alpha g$ for some $g$ so $\beta \in \Gamma g$ and the blocks cover $\Omega$. If $\Gamma g \cap \Gamma h$ is non-empty then $\Gamma gh^{-1} = \Gamma$ so $\Gamma g = \Gamma h$.

Definition A $G$-congruence is an equivalence relation $R$ on $\Omega$ such that $\alpha R \beta$ implies $\alpha g R \beta g$.

Definition If $R$ is a $G$-congruence then the $R$-equiv classes form a block system and conversely if $\Gamma$ is a block we define $R$ by $\alpha R \beta$ iff $\alpha,\beta \in \Gamma g$.
proof: easy

The trivial $G$-congruences are the equality relation and the one induced by the block $\Omega$.

Definition The (transitive) action on $\Omega$ is called imprimitive if there is a non-trivial $G$-congruence. If there are no non-trivial $G$-congruences an action is primitive.

Proposition Let $\alpha \in \Omega$, write $B(\alpha)$ for the set of blocks containing $\alpha$ and $S(\alpha)$ for the set of subgroups containing $G_\alpha$.

• There are mutually inverse bijections $\Psi : B(\alpha) \to S(\alpha)$ and $\Phi : S(\alpha) \to B(\alpha)$ defined by $\Gamma \Psi = G_{\Gamma}$, $H \Phi = \alpha H$.
• For $\Gamma,\Gamma' \in B(\alpha)$ we have $\Gamma \subseteq \Gamma'$ iff $\Gamma \Psi \le \Gamma' \Psi$.

proof: long.

Corollary The action of $G$ on $\Omega$ is primitive iff each $G_\alpha$ is a maximal subgroup.

Proposition If the action of $G$ on $\Omega$ is 2-transitive then it is primitive.
proof: If the action is 2-trans take $\alpha \in \Omega$. Suppose $\Gamma$ is a block containing $\alpha$ with $|\Gamma| > 1$. Take $\beta \in \Gamma \setminus \{\alpha\}$ then for any $\beta' \in \Omega \setminus \{\alpha\}$ there exists $g \in G_\alpha$ with $\beta g = \beta'$. As $\alpha \in \Gamma g \cap \Gamma$ we must have $\Gamma g = \Gamma$ so $\beta' \in \Gamma$ and hence $\Gamma = \Omega$. So $\alpha$ lies in no nontrivial block.

Note: The converse is not true, you can have imprimitive actions that aren't 2-transitive.

Proposition If $N \unlhd G$ the set of $N$-orbits in $\Omega$ is a block system.
proof:  Let $\Gamma$ be an $N$-orbit, If $g \in G$ with $\Gamma g \cap \Gamma \not = \{\}$ let $\alpha \in \Gamma g \cap \Gamma$ so $\alpha = \beta g$ with $\beta \in \Gamma$ then $\Gamma = \alpha N = \beta N$. So $\Gamma g = \beta N g = \beta g N = \alpha N = \Gamma$.

Corollary If $G$ is a primitive permutation group (no non-identity elements fix all elements of $\Omega$) and $1 \not = N \unlhd G$ then $N$ acts transitively.
proof: Since $1 \not = N$ there is an $N$ orbit of size > 1 so by the previous theorem it gives a block system, by primitivity it's the whole of $\Omega$, so $N$ must act transitively.