While I am reading Serge Lang's Algebra, I am confused by the definition of solvable group (page 18).
In the book, $G$ is solvable if $G$ has an abelian tower with last element being trivial subgroup. Is he trying to say a group with an abelian tower may be non-solvable? If so, what is an example of this?
It seems to me that if $G$ has an abelian tower then $G$ is solvable: say $$ G_{1}\subset G_2 \subset \cdots \subset G_n=G $$ is an abelian tower of $G$. Why can't I just add $\{e\}=G_{0}$ (which is clearly normal in $G_{1}$) to the sequence? To me it looks like possessing an abelian tower implies solvability.