The Influence of $s$-Conditional Permutability of Subgroups on the Structure of Finite Groups
Received:May 31, 2009  Revised:September 18, 2009
Key Word: finite groups   $s$-conditionally permutable groups   saturated formations   supersoluble groups   nilpotent groups.  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11071229), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No.10KJD110004) and the Postgraduate Innovation Grant of Xuzhou Normal University.
Author NameAffiliation
Wen Juan NIU Department of Mathematics, Xuzhou Normal University, Jiangsu 221116, P. R. China 
Wen Bin GUO Department of Mathematics, Xuzhou Normal University, Jiangsu 221116, P. R. China
Department of Mathematics, University of Science and Technology of China, Anhui 230026, P. R. China 
Yu Feng LIU School of Mathematical and Informational Science, Shandong Institute of Business and Technology, Shandong 264005, P. R. China 
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      Let $G$ be a finite group. Fix a prime divisor $p$ of $|G|$ and a Sylow $p$-subgroup $P$ of $G$, let $d$ be the smallest generator number of $P$ and ${\cal M}_{d}(P)$ denote a family of maximal subgroups $P_{1}, P_{2}, \ldots, P_{d}$ of $P$ satisfying $\bigcap_{i=1}^{d}P_{i}=\Phi(P)$, the Frattini subgroup of $P$. In this paper, we shall investigate the influence of $s$-conditional permutability of the members of some fixed ${\cal M}_{d}(P)$ on the structure of finite groups. Some new results are obtained and some known results are generalized.
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