Two Frattini Embedding Theorems for Infinite Nilpotent Groups

In this paper, two Frattini embedding theorems on nilpotent groups are proved. First, let [Formula: see text] be a finitely generated nilpotent group whose torsion subgroup is a finite [Formula: see text]-group, and then there exists a finitely generated nilpotent group [Formula: see text] whose torsion subgroup is also a finite [Formula: see text]-group such that [Formula: see text]. Second, let [Formula: see text] be a nilpotent Cernikov [Formula: see text]-group for a finite set [Formula: see text] of primes, and then there exists a nilpotent Cernikov [Formula: see text]-group [Formula: see text] such that [Formula: see text] and [Formula: see text].