INFINITE GROUPS WITH TWO CONJUGACY CLASSES OF NON-SUBNORMAL SUBGROUPS

Let G be an infinite group. mu denotes the number of the conjugacy classes of non-subnormal subgroups of G. mu(<infinity) and mu(infinity) denote the number of the finitely length and infinitely length conjugacy classes of non-subnormal subgroups of G, respectively. Let G be a group with mu = 2. Then it is proved that (1) there exists no infinite group with mu(<infinity) = 2; (2) if mu(<infinity) = mu(infinity) = 1, then there exists some normal subgroup N angle G, such that G/N is a finite non-nilpotent inner-abelian group, where N is a group with all subgroups subnormal; (3) if G is infinite locally finite, then G is a Baer group, and mu(infinity) = 2.