# primitive permutation group

Let $X$ be a set, and $G$ a transitive permutation group on $X$. Then $G$ is said to be a primitive permutation group if it has no nontrivial blocks (http://planetmath.org/BlockSystem).

For example, the symmetric group $S_{4}$ is a primitive permutation group on $\{1,2,3,4\}$.

Note that $D_{8}$ is not a primitive permutation group on the vertices of a square, because the pairs of opposite points form a nontrivial block.

It can be shown that a transitive permutation group $G$ on a set $X$ is primitive if and only if the stabilizer $\operatorname{Stab}_{G}(x)$ is a maximal subgroup of $G$ for all $x\in X$.

Title primitive permutation group PrimitivePermutationGroup 2013-03-22 14:00:49 2013-03-22 14:00:49 Thomas Heye (1234) Thomas Heye (1234) 20 Thomas Heye (1234) Definition msc 20B15