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 $G$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Jones on 2018-07-07 20:33:52

**Referred to by:**

**History:**(expand/hide all)

- 2018-07-07 20:33:52 by John Jones (Reviewed)