# transitive actions are primitive if and only if stabilizers are maximal subgroups

###### Theorem 1.

If $G$ is transitive on the set $A$, then $G$ is primitive on $A$ if and only if for each $a\in A$, $G_{a}$ is a maximal subgroup of $G$. Here $G_{a}=\operatorname{Stab}_{G}(a)$ is the stabilizer of $a\in A$.

###### Proof.

First claim that if $G$ is transitive on $A$ and $B\subset A$ is a block (http://planetmath.org/BlockSystem) with $a\in B$, then $G_{B}=\{\sigma\in G\ \mid\ \sigma(B)=B\}$ is a subgroup of $G$ containing $G_{a}$. It is obvious that $G_{B}$ is a subgroup, since

 $\displaystyle\sigma\in G_{B}\Rightarrow\sigma(B)=B\Rightarrow\sigma^{-1}(% \sigma(B))=\sigma^{-1}(B)\Rightarrow B=\sigma^{-1}(B)\Rightarrow\sigma^{-1}\in G% _{B}$ $\displaystyle\sigma,\tau\in G_{B}\Rightarrow(\sigma\tau)(B)=\sigma(\tau(B))=% \sigma(B)=B\Rightarrow\sigma\tau\in G_{B}$

But also, if $\sigma\in G_{a}$ for $a\in B$, then $\sigma(a)=a$, so $\sigma(B)\cap B\neq\emptyset$ and thus $\sigma(B)=B$ since $B$ is a block system and thus $\sigma\in G_{B}$. This proves the claim.

To prove the theorem, note that for each $a\in A$, there is by the claim a $1-1$ correspondence between containing $a$ and subgroups of $G$ containing $G_{a}$. Thus, $G$ is primitive on $A$ if and only if all blocks are either of size $1$ or equal to $A$, if and only if any group containing $G_{a}$ is either $G_{a}$ itself or $G$, if and only if for all $a\in A$, $G_{a}$ is maximal in $G$. ∎

Title transitive actions are primitive if and only if stabilizers are maximal subgroups TransitiveActionsArePrimitiveIfAndOnlyIfStabilizersAreMaximalSubgroups 2013-03-22 17:19:07 2013-03-22 17:19:07 rm50 (10146) rm50 (10146) 6 rm50 (10146) Theorem msc 20B15