A Classification of Minimal Non-MNP-Groups

Abstract

A finite group G is called an MNP-group if all maximal subgroups of every Sylow subgroup of G are normal in G. In this article, we give a complete classification of those groups which are not MNP-groups but all of whose proper subgroups are MNP-groups.

Key Words:

• MNP-groups
• Minimal non-MNP-groups
• Power automorphisms
• Supersolvable groups

2000 Mathematics Subject Classification:

• 20D10
• 20D20

ACKNOWLEDGEMENTS

The authors express their deep sense for gratitude to the anonymous referee for his/her valuable suggestions and comments which polish largely this article. The authors also thank Professor Junxin Wang for his helpful suggestions.

The research of the work was supported by the National Natural Science Foundation of China (No. 11071155, No. 11201401), China (No. 20100480582), and the Overseas Study Project of Jiangsu Province Jiangsu Overseas Research & Training Program for Outstanding Young and Middle-aged Teachers and Principals (No. 2011[34]), and Qing Lan Project of Jiangsu Province.

Notes

Communicated by A. Olshanskii.