Geodesic
On the intersection of maximal supersoluble subgroups of a finite group
Trudy Instituta matematiki, Tome 21 (2013) no. 1, pp. 48-51

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The hyper-generalized-center $genz^*(G)$ of a finite group $G$ coincides with the largest term of the chain of subgroups $1=Q_0(G)\le Q_1(G)\le\ldots\le Q_t(G)\le\ldots$ where $Q_i(G)/Q_{i-1}(G)$ is the subgroup of $G/Q_{i-1}(G)$ generated by the set of all cyclic $S$-quasinormal subgroups of $G/Q_{i-1}(G)$. It is proved that for any finite group $A,$ there is a finite group $G$ such that $A\le G$ and $genz^*(G)\ne\text{Int}_\mathfrak{U}(G)$. 
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Wenbin Guo; Alexander N. Skiba. On the intersection of maximal supersoluble subgroups of a finite group. Trudy Instituta matematiki, Tome 21 (2013) no. 1, pp. 48-51. http://geodesic.mathdoc.fr/item/TIMB_2013_21_1_a5/