Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

Khaled A. Al-Sharo [a] and Abdulla A. Sharo [b]

On $$m$$-$$S$$-complemented subgroups of finite groups

Pages: 351-363
Accepted in revised form: 24 January 2019
Mathematics Subject Classification (2010): 20D10, 20D15, 20D30.
Keywords: Finite group, modular subgroup, $$S$$-quasinormal subgroup, generalized $$S$$-quasinormal subgroup, $$m$$-$$S$$-complemented subgroup.
[a]: Dept. of Mathematics, Al al-Bayt University, Mafraq-25113, Jordan.
[b]: Dept. of Civil Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan.

Abstract: Let $$G$$ be a finite group and $$H$$ a subgroup of $$G$$. We say that $$H$$: is generalized $$S$$-quasinormal in $$G$$ if $$H=\langle A, B \rangle$$ for some modular subgroup $$A$$ and $$S$$-quasinormal subgroup $$B$$ of $$G$$; $$m$$-$$S$$-complemented in $$G$$ if there are a generalized $$S$$-quasinormal subgroup $$S$$ and a subgroup $$T$$ of $$G$$ such that $$G=HT$$ and $$H\cap T\leq S\leq H$$. In this paper, we study finite groups with given systems of $$m$$-$$S$$-complemented subgroups. In particular, we prove the following result: Let $$\mathfrak{F}$$ be a saturated formation containing all supersoluble groups, and let $$E$$ be a normal subgroup of a finite group $$G$$ such that $$G/E\in \mathfrak{F}$$. If for any Sylow subgroup $$P$$ of $$E$$ every maximal subgroup of $$P$$ not having a supersoluble supplement in $$G$$ is $$m$$-$$S$$-complemented in $$G$$, then $$G\in \mathfrak{F}$$.

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