About groups with property $(U)$ - Rostami, Hamid Reza (Armenian Journal of Mathematics, 2008)
In this article we prove that for any odd $n \ge 1003$ there exist continuum non isomorphic (simple) groups with property $(U)$, whose $U$- constant satisfies the inequality $u(G) < (3n^4)/2$. The question of finding infinite groups with property $(U)$ was proposed in the joint work by D.Osin and D.Sonkin.