On generalizations of Iwasawa's theorem

2024/03/13 by Shi, Jiangtao, Xu, Fanjie, Shan, Mengjiao
#20D10 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.17960

Abstract

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.

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