A note on Huppert's theorem and Chen's theorem

Let G be a finite group, we prove that every maximal subgroup of G has prime index if and only if every maximal subgroup of G that contains the normalizer of some Sylow subgroup has prime index, which implies that the hypothesis in Huppert’s theorem and the hypothesis in Chen’s theorem are actually equivalent. Moreover, we prove that the hypothesis in a theorem of Shao and Beltrán and the hypothesis in a theorem of Li et al are also equivalent.