The nilpotency of some groups with all subgroups subnormal (Q1281290)

Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\) is finitely generated for each \(x\in G\) then \(G\) is nilpotent [\textit{H. Smith}, Glasg. Math. J. 36, No. 1, 33-36 (1994; Zbl 0803.20019)]. The article under review is dedicated to the following question: what other conditions on \(\langle x\rangle^G\) imply the nilpotency of \(G\)? Let \(G\) be a group, \(H\) a normal subgroup of \(G\). Then \(H\) is said to be \(G\)-minimax, if \(H\) has a finite series of \(G\)-invariant subgroups every factor of which is abelian and satisfies Max-\(G\) or Min-\(G\). The main results of this paper are the following theorems. Theorem A. Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\) is \(G\)-minimax for all \(x\in G\) then (1) \(G\) is nilpotent; (2) \(\langle x\rangle^G\) is minimax for all \(x\in G\). Theorem B. Let \(G\) be a group with all subgroups subnormal. If \(\langle x\rangle^G\in{\mathbf S}_1\) then \(G\) is nilpotent. Corollary. Let \(G\) be a group with all subgroups subnormal. If \(G\) has minimax conjugacy classes then \(G\) is nilpotent.