Year: 2026
Type: preprint
Abstract: Let $p$ be a prime number. When $p$ is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or $p$-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or $p$-decomposable. We prove that every non-simple, non-solvable group satisfying the first condition is $p$-nilpotent, and every non-simple, non-solvable group satisfying the second condition is $p$-decomposable. In addition, we determine the possible occurrence of non-abelian simple maximal subgroups in these cases. These results provide a substantial partial answer to two questions posed by V.S. Monakhov and I.N. Tyutyanov in the Kourovka Notebook concerning the non-abelian composition factors of such groups. For non-abelian simple groups, we determine those satisfying the corresponding conditions within the alternating and sporadic families. The case of simple groups of Lie type is left open. Finally, for $p=2$, we obtain a complete classification of the non-solvable finite groups whose maximal subgroups are either non-abelian simple or $2$-nilpotent.
Source: arXiv (Cornell University)
Authors: Beltrán, Antonio, Viudez, Fernando
Institution: Universitat Jaume I
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Universitat Jaume I
OpenAlex ID: W7167731114