The influence of complemented minimal subgroups on the structure of finite groups

• Autores: Jiakuan Lu, Shirong Li, Xiuyun Guo
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 76, Fasc. 3-4, 2010, págs. 481-491
• Idioma: inglés
• DOI: 10.5486/pmd.2010.4586
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• Resumen

  • A subgroup H of a ¯nite group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H \ K = 1. In this paper the following theorem is proved: Let G be a ¯nite group and let p be the smallest prime dividing the order of G. Then G is p-nilpotent if and only if every minimal subgroup of P \ GN is complemented in NG(P), where P is a Sylow p-subgroup of G and GN is the nilpotent residual of G. As some applications, some interesting results related with complemented minimal subgroups are obtained.