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...

Let $\lambda$ be an infinite cardinal. Set ${\lambda }_0=\lambda$, define ${\lambda }_{i+1}=2^{{\lambda }_i}$ for every $i=0,1,\cdots$, take $\mu$ as the first cardinal with ${\lambda }_i<\mu$, $i=0,1,\cdots$ and put $\kappa ={\left({\mu }^{{\aleph }_0}\right)}^{+}$. If $F$ is a torsion-free group of cardinality at least $\kappa$ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda$, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$$p$-primary.