Infinite groups with one conjugate class of non-subnormal subgroups. (Q2862003)

The authors show how much the possibilities for groups of the title are restricted: If \(G\) is a group of the title, then the following statements are equivalent: (i) \(G\) is locally nilpotent, (ii) \(G\) satisfies the normalizer condition, (iii) \(G\) is a Baer group. (Theorem 1). Further \(G\) cannot be locally finite (Theorem 3). If all proper subgroups of \(G\) are finite, then \(G/Z(G)\) is a Tarski \(p\)-group (Theorem 4).