Primitive permutation group

In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive. In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive. While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when X is a 2-element set and the action is trivial; otherwise, the condition that G preserves no nontrivial partition implies that G is transitive. This is because for non-transitive actions either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case any nontrivial partition of X (which exists for |X|≥3) is preserved by G. This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive.