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 $x\\in X$ .

单词	PrimitivePermutationGroup
释义	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 $x\\in X$ .
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