Volume 34 • Number 3 • 2011   • New Characterizations of Some Classes of Finite Groups Wenbin Guo, Xiuxian Feng and Jianhong Huang Abstract. Let $G$ be a finite group and $\frak{F}$ a formation of finite groups. We say that a subgroup $H$ of $G$ is $\mathfrak{F}_h$-normal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a normal Hall subgroup of $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty ^\frak{F} (G/H_G)$ of $G/H_G$. In this paper, we obtain some results about the $\mathfrak{F}_h$-normal subgroups and use them to study the structure of finite groups. Some new characterizations of supersoluble groups, soluble groups and $p$-nilpotent groups are obtained and some known results are generalized. 2010 Mathematics Subject Classification: 20D10, 20D15, 20D20. Full text: PDF