Groups whose subgroups satisfy the weak subnormalizer condition (Q2334475)

A subgroup \(X\) of a group \(G\) is said to satisfy the \textit{weak subnormalizer condition} if \(N_G(Y) \leq N_G(X)\) for each non-normal subgroup \(Y\) of \(G\) such that \(X \leq Y \leq N_G(X)\). In the following, \(\mathcal{W}\) (respectively \(\mathcal{W}_c\)) will indicate the class of groups in which every (cyclic) subgroup satisfies the weak subnormalizer condition. To avoid pathological situations like Tarski groups, the authors prove their results within a suitable universe of generalized soluble groups. Under the hypothesis of local nilpotency, the authors study \(\mathcal{W}_c\)-groups proving that such groups are nilpotent of class at most \(3\) and even abelian if they are torsion-free. The consideration of the quaternion group of order \(16\) shows that there are \(W_c\)-groups of nilpotency class \(3\), so the result is sharp. Continuing the analysis of \(\mathcal{W}_c\)-groups, they prove that locally finite \(\mathcal{W}_c\)-groups are soluble of derived length at most \(4\): it is an open question if the bound on the derived length is sharp. Finally, the authors prove that a \(W\)-group \(G\) is soluble of derived length at most \(5\) provided that \(G\) is \textit{weakly radical} (i.e. it admits an ascending normal series whose factors are either locally soluble or locally finite).