New Conditions on Maximal Invariant Subgroups That Imply Solubility

Let G be a finite group and assume that a finite group of automorphisms A acts on G , such that the orders of A and G are relatively prime. We prove that the fact of imposing certain conditions on the set of maximal A -invariant subgroups of G , relating to nilpotency, p -nilpotency, normality or having p’ -order, determines properties on the structure of G such as solubility, p -solubility or p -nilpotency.