Primitive permutation group

Summary

A permutation group is primitive when it is transitive and preserves no nontrivial block system, that is, no nontrivial splitting of the set into parts. Primitive groups are the building blocks of all finite permutation groups. A transitive group of degree $n$ that is not primitive embeds in a wreath product $S_k \wr S_m$ with $km = n$ and $1 < k, m < n$. A transitive group is primitive exactly when its point stabilisers are maximal subgroups. The O'Nan–Scott theorem sorts the finite primitive groups into a few types.