On the structure of a group whose orbits on a finite module are the orbits of a proper subgroup
The authors prove a result they call the Orbit Theorem, which has been used by \textit{T. Hawkes} and \textit{J. Humphreys} [J. Algebra 94, 382-387 (1985; Zbl 0594.20007)] to give a criterion for existence of normal complements. Let p, q be distinct primes, let G be a finite solvable group and let Q be a normal elementary abelian subgroup of G, complemented in G by a subgroup H. Let V be a kG-module, faithful for Q, in which the G-orbit of each element coincides with its H-orbit. The Orbit Theorem asserts that under these conditions, \([Q,O_{q'}(H)]=1\).
scientific article; zbMATH DE number 3968870
scientific article