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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 $G$.

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