Geodesic
On $\mathcal{U}\Phi$-hypercentre of finite groups
Problemy fiziki, matematiki i tehniki, no. 1 (2010), pp. 28-30

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The product of all normal subgroups of $G$ whose all non-Frattini $G$-chief factors are cyclic is called the $\mathcal{U}\Phi$-hypercentre of $G$. The following theorem is proved.Theorem. Let $X \le E$ be soluble normal subgroups of $G$. Suppose that every maximal subgroup of every Sylow subgroup of $X$ conditionally covers or avoids each maximal pair $(M,G)$, where $MX = G$. If $X$ is either $E$ or $F(E)$, then. $E \le Z_{\mathcal{U}\Phi}(G)$.
Keywords: $\mathcal{U}\Phi$-hypercentre, supersoluble group, (conditionally) cover-avoidance property of subgroups, CAP-subgroup.
Mots-clés : maximal pair 
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V. A. Kovaleva; A. N. Skiba. On $\mathcal{U}\Phi$-hypercentre of finite groups. Problemy fiziki, matematiki i tehniki, no. 1 (2010), pp. 28-30. http://geodesic.mathdoc.fr/item/PFMT_2010_1_a5/