Now that school is wrapping up, I'm trying to crack down and get better at algebra. This proposition from Lang's Algebra loses me at the end.
Here is my understanding so far: (Please excuse me if a lot of the things I say are very obvious/wrong, I'm trying to be detailed for my own understanding.) If $|G|=1$, then $\{e\}$ is the desired cyclic tower. So suppose the result holds for $|G|\leq n-1$. Suppose $|G|=n$. Letting $G'$ be as above, $|G'|=|G|/|X|\lt|G|$, so by the induction hypothesis, there exists a cyclic tower in $G'$, say $ G'=G/X\supset G_1'\supset G_2'\supset\cdots\supset G_m'. $ I'm not quite sure what Lang means by "its inverse image is a cyclic tower in $G$ whose last element is $X$." Is there some assumed homomorphism $f\colon G\to G'$, and then the inverse image of the tower would be $ f^{-1}(G')\supset f^{-1}(G_1')\supset\cdots\supset f^{-1}(G_m')? $ Why is the last element of the tower $X$, as Lang claims? Also, Lang says a normal tower is cyclic if each factor group $G_i/G_{i+1}$ in the tower is cyclic. Does this mean all the $G_i$ themselves are cyclic, or is it possible for the factor group to be cyclic, but the normal subgroup being modded out is not? Thanks for any explanation.