On CISE-normal subgroups of finite groups. (Q2911321)

Let \(G\) be a finite group and \(H\) a subgroup of \(G\). \textit{O.~H.~Kegel} [Math. Z. 78, 205-221 (1962; Zbl 0102.26802)] called \(H\) \(s\)-quasinormal in \(G\) if \(PH=HP\) for all Sylow subgroups \(P\) of \(G\). \textit{A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera} [J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)] called \(H\) \(\pi\)-quasinormally embedded in \(G\) if for each prime number \(p\in\pi(H)\), a Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(s\)-quasinormal subgroup of \(G\). \textit{H. Wei} and \textit{Y. Wang} [J. Group Theory 10, No. 2, 211-223 (2007; Zbl 1125.20011)] called \(H\) \(c^*\)-normal in \(G\) if \(G\) has a normal subgroup \(T\) such that \(G=HT\) and \(H\cap T\) is \(\pi\)-quasinormally embedded in \(G\). If there exists a subnormal subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\) is \(\pi\)-quasinormally embedded in \(G\), then \(H\) is called an ISE-normal subgroup of \(G\). If \(H\) has a supersoluble supplement in \(G\) or \(H\) is ISE-normal in \(G\), then \(H\) is called a CISE-normal subgroup of \(G\).NEWLINENEWLINE The main results are as follows.NEWLINENEWLINE Theorem 3.1: If \(\mathcal F\) is a saturated formation containing the class \(\mathcal U\) of supersoluble groups, \(G\) has a normal subgroup \(N\) such that \(G/N\in\mathcal F\) and every Sylow subgroup of \(F^*(N)\) is cyclic, then \(G\in\mathcal F\).NEWLINENEWLINE Theorem 3.2: If \(\mathcal F\) is a saturated formation containing \(\mathcal U\), \(G\) has a normal subgroup \(N\) such that \(G/N\in\mathcal F\) and every non-cyclic Sylow subgroup \(P\) of \(N\) has a subgroup \(U\) with \(1<|U|<|P|\) such that every subgroup \(H\) of \(P\) of order \(|U|\) and every cyclic subgroup of \(P\) of order 4 (if \(|U|=2\) and \(P\) is a non-Abelian 2-group) is CISE-normal in \(G\), then \(G\in\mathcal F\).NEWLINENEWLINE Theorem 3.4: If \(\mathcal F\) is a saturated formation containing \(\mathcal U\), \(G\) has a normal subgroup \(N\) such that \(G/N\in\mathcal F\) and every non-cyclic Sylow subgroup \(P\) of \(F^*(N)\) has a subgroup \(U\) with \(1<|U|<|P|\) such that every subgroup \(H\) of \(P\) of order \(|U|\) and every cyclic subgroup of \(P\) of order 4 (if \(|U|=2\) and \(P\) is a non-Abelian 2-group) is CISE-normal in \(G\), then \(G\in\mathcal F\).