In order to give a further study of the solvability of a finite group in which every non-nilpotent maximal subgroup has prime index

the methods of the proof by contradiction and the counterexample of the smallest order and a theorem of Wielandt on the characterization of the structure of a finite group G with a nilpotent Hall-subgroup which is not a Sylow subgroup are applied to obtain a more elementary proof of the solvability of a finite group in which every non-nilpotent maximal subgroup has prime index. The proof does not apply the Glauberman-Thompson p-nilpotent criterion and Rose′s two results on a classification of non-abelian simple groups with nilpotent maximal subgroup and a characterization of non-solvable group with nilpotent maximal subgroup and trivial center respectively

which improves the proof of the result in the relevant research references.