# doubly transitive groups are primitive

###### Theorem.

Every doubly transitive group is primitive (http://planetmath.org/PrimativeTransitivePermutationGroupOnASet).

###### Proof.

Let $G$ acting on $X$ be doubly transitive. To show the action is , we must show that all blocks are trivial blocks; to do this, it suffices to show that any block containing more than one element is all of $X$. So choose a block $Y$ with two distinct elements $y_{1},y_{2}$. Given an arbitrary $x\in X$, since $G$ is doubly transitive, we can choose $\sigma\in G$ such that

 $\sigma\cdot(y_{1},y_{2})=(y_{1},x)$

But then $\sigma\cdot Y\cap Y\neq\emptyset$, since $y_{1}$ is in both. Thus $\sigma\cdot Y=Y$, so $x\in Y$ as well. So $Y=X$ and we are done. ∎

Title doubly transitive groups are primitive DoublyTransitiveGroupsArePrimitive 2013-03-22 17:21:50 2013-03-22 17:21:50 rm50 (10146) rm50 (10146) 7 rm50 (10146) Theorem msc 20B15