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 H^G = G, where H^G 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 ۰-rank of A is finite, the set \Pi(A) is finite, G/A is a Baer group, and every subnormal subgroup of G is contranormal-free, then G is a Baer group.

Keywords:

contranormal subgroups , subnormal subgroups , Nilpotent groups , hypercentral groups , upper central series