Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018
Khaled A. AlSharo ^{[a]}
and
Abdulla A. Sharo ^{[b]}
On \(m\)\(S\)complemented subgroups of finite groups
Pages: 351363
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 alBayt University, Mafraq25113, Jordan.
[b]: Dept. of Civil Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan.
Full Text (PDF)
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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