Cyclic quotients of transitive groups
In this note two interesting theorems about finite permutation groups and an application to the Galois theory of function fields resp. algebraic curves are proved. Theorem 1. If \(A\) is a transitive subgroup of the symmetric group \(S_n\) on \(n\) letters and \(G\) is a normal subgroup of \(A\) such that \(A/G\) is cyclic, then \(|A/G|\leq n\) and \(|A/G|<n\) if \(G\) is transitive. Theorem 3. If \(A\) is a finite primitive permutation group then a point stabilizer in \(A\) has a faithful orbit. The formulation of the mentioned application is more complicated (see Theorem 2 and sections 5 and 6 of the paper). The proofs are based on using a new group invariant \(\mu(H)\) related to central chief factors in normal series of the group \(H\). The classification of finite simple groups is used for obtaining the full results. However, some partial results are proved without invoking the classification.
scientific article; zbMATH DE number 1565872
scientific article