This article is dedicated to some criteria of generalized nilpotency involving pronormality and abnormality. Also new results on groups, in which abnormality is a transitive relation, have been obtained.

In this paper, we give a generalization of Baer Theorem on the injective property of divisible abelian groups. As consequences of the obtained result we find a sufficient condition for a group $G$ to express as semi-direct product of a divisible subgroup $D$ and some subgroup $H$. We also apply the main Theorem to the $p$-groups with center of index $p^2$, for some prime $p$. For these groups we compute $N_c\left(G\right)$ the number of conjugacy classes and $N_a$ the number of abelian maximal subgroups and $N_{na}$ the number of nonabelian...