On nearly ${S\Phi}$-normal subgroups of finite groups

M. T. Hussain, S. Ullah

Let

$G$ be a finite group,

$H$ a subgroup of

$G$ and

$H_{sG}$ the subgroup of

$H$ generated by all those subgroups of

$H$ which are

$s$ -permutable in

$G$ . Then we say that

$H$ is

$\textit{nearly $S\Phi$-normal}$ in

$G$ if

$G$ has a normal subgroup

$T$ such that

$HT\unlhd G$ and

$H\cap T\leq \Phi (H)H_{sG}$ . In this paper, we study the structure of group

$G$ under the condition that some given subgroups of

$G$ are nearly

$S\Phi$ -normal in

$G$ . Some known results are generalised.

finite group, nearly

$S\Phi$ -normal subgroup, Sylow

$p$ -subgroup,

$p$ -nilpotent group, supersoluble group

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DOI: http://dx.doi.org/10.12958/adm2007