Finite groups with simple or p-nilpotent maximal subgroups
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.
Related
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- On second maximal subgroups of finite groups
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BibTeX
@misc{beltran2026finite,
title = {{Finite groups with simple or $p$-nilpotent maximal subgroups}},
author = {Antonio Beltrán and Fernando Viudez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.06434},
eprint = {2607.06434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.06434},
}
RIS
TY - UNPB
TI - Finite groups with simple or p-nilpotent maximal subgroups
AU - Antonio Beltrán
AU - Fernando Viudez
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2607.06434
UR - https://arxiv.org/abs/2607.06434
AB - 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.
ER -