On some properties of pronormal subgroups. (Q539179)

Let \(H\) be a subgroup of a group \(G\). The subgroup \(H\) is called abnormal in \(G\) if for each element \(g\in G\) we have \(g\in\langle H,H^g\rangle\). The subgroup \(H\) is called pronormal in \(G\) if for each element \(g\in G\) the subgroups \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\). The subgroup \(H\) is called weakly pronormal in \(G\) if for any two intermediate subgroups \(K,L\) for \(H\) such that \(K\) is normal in \(L\) the following inclusion holds: \(L\leq N_G(H)K\). The subgroup \(H\) is called nearly pronormal if \(N_K(H)\) is contranormal in \(K\) for every \(K\geq H\). The subgroup \(H\) is called contranormal, if the normal closure \(H^G\) coincides with the whole group \(G\). Abnormal subgroups are contranormal but the converse is not valid in general. A group is called \(N\)-group if it satisfies the normalizer condition, that is, every proper subgroup of the group is properly contained in its normalizer in the group. The authors of the paper under review have established tight connections among pronormal, abnormal and contranormal subgroups of a group. The main results of the paper under review are the following. Theorem 1.1. Let \(G\) be a hyper-\(N\)-group. Then every nearly pronormal subgroup is pronormal in \(G\). A group \(G\) is called an \(\widetilde N\)-group if \(G\) satisfies the following condition: if \(M\) and \(L\) are subgroups of \(G\) such that \(M\) is maximal in \(L\), then \(M\) is normal in \(L\). Theorem 1.5. Let \(G\) be a hyper-\(\widetilde N\)-group. Then every nearly abnormal subgroup of \(G\) is abnormal in \(G\).