Primitivity of a Galois group (reviewed)

Let $G$ be a transitive subgroup of $S_n$. Then $G$ is primitive if, under the Galois correspondence, it corresponds to a field with no non-trivial proper subfields.

An equivalent group-theoretic condition is as follows. Let \[ G_1 = \{\sigma\in G\mid \sigma(1)=1\}.\] Then $G$ is primitive if and only if $G_1$ is a maximal subgroup of $G$.