Excerpt:
Discussion:
I think I understand every part of this except for the cyclic part, and the part where he refers to subgroups at the end. I will write my understanding of everything else to confirm that I"m on the right track.
The $G_i$ form a normal tower in $G$ because the homomorphism preimage of a normal subgroup in $G'$ is normal in $G$. The injective homomorphism comes about because of a triangular diagram I don't know how to draw here, but it would have this shape:
with $G$ replaced with $G_i$, $G'$ replaced with $G_i'/G_{i + 1}'$, and $G/N$ replaced with $G_i / G_{i + 1}$. The arrow on the right is the injective homomorphism described by Lang.
If the $G_i'$ form an abelian tower, then the $G_i$ form an abelian tower because an injective homomorphism into an abelian group implies that the domain group is abelian (one-line proof).
I'm not sure how to show that a cyclic tower $G_i'$ implies that $G_i$ is a cyclic tower. Am I supposed to use the injective homomorphism? And which subgroups is Lang referring to in the last two lines?
I appreciate any help.
Edit:
Is it that I could turn the injective homomorphism into an isomorphism by reducing the codomain, and then if two groups (i.e. the original domain and the image of the homomorphism in the codomain) are isomorphic and one is cyclic, then the other one is as well?