On some generalization of the malnormal subgroups

A subgroup H of a group G is called malonormal in G if H ∩H = ⟨1⟩ for every element x / ∈ NG(H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every selfnormalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.