Geodesic
Characterization of finite groups with some $S$-quasinormal subgroups of fixed order
Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 161-167

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Let $G$ be a finite group. A subgroup of $G$ is said to be $S$-quasinormal in $G$ if it permutes with every Sylow subgroup of $G$. We fix in every non-cyclic Sylow subgroup $P$ of the generalized Fitting subgroup a subgroup $D$ such that $1 |D| |P|$ and characterize $G$ under the assumption that all subgroups $H$ of $P$ with $|H| = |D|$ are $S$-quasinormal in $G$. Some recent results are generalized.
Keywords: $S$-quasinormality, generalized Fitting subgroup, supersolvability. 
@article{ADM_2012_14_2_a1,
     author = {M. Asaad and Piroska Cs\"org\H{o}},
     title = {Characterization of finite groups with some $S$-quasinormal subgroups of fixed order},
     journal = {Algebra and discrete mathematics},
     pages = {161--167},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2012_14_2_a1/}
}
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M. Asaad; Piroska Csörgő. Characterization of finite groups with some $S$-quasinormal subgroups of fixed order. Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 161-167. http://geodesic.mathdoc.fr/item/ADM_2012_14_2_a1/