On generalizations of Iwasawa's theorem

Iwasawa's theorem indicates that a finite group G is supersolvable if and only if all maximal chains of the identity in G have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group G in which all maximal chains of every minimal subgroup have the same length. Moreover, let δ(G) be the number of subgroups of G all of whose maximal chains in G do not have the same length, we prove that G is a non-solvable group with δ(G)≤ 16 if and only if G A₅.