on some groups whose subnormal subgroups are contranormal-free

If g is a group, a subgroup h of g is said to be contranormal in g if hg = g, where hg is the normal closure of h in g. we say that a group is contranormal-free if it does not contain proper contranormal subgroups. obviously, a nilpotent group is contranormal-free. conversely, if g is a finite contranormal-free group, then g is nilpotent. we study (infinite) groups whose subnormal subgroups are contranormal-free. we prove that if g is a group which contains a normal nilpotent subgroup a such that g/a is a periodic baer group, and every subnormal subgroup of g is contranormal-free, then g is generated by subnormal nilpotent subgroups; in particular g is a baer group. furthermore, if g is a group which contains a normal nilpotent subgroup a such that the 0-rank of a is finite, the set π(a) is finite, g/a is a baer group, and every subnormal subgroup of g is contranormal-free, then g is a baer group.