Details

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

On \(m\)-\(S\)-complemented subgroups of finite groups

Pages: 351-363
Received: 23 October 2018
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.
Authors address:
[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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