Invariant Sylow subgroups and solvability of finite groups

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.This research is supported by Universitat Jaume I, Grant P11B2012-05, and by the Valencian Government, Proyecto PROMETEOII/2015/011