Is the Solution for Problem 2 on the Algebra Qualifier Exam Correct?

[mrbohn1]

I'm looking at solutions to an algebra qualifying exam someone has posted on the web; the page is here:

http://mathwiki.gc.cuny.edu/index.php/Spring_2007_Algebra_Qualifier"

I'm looking at problem 2.

Is this solution OK? The author has not addressed the necessary condition that the solvable series has cyclic quotient groups, and I'm not sure that the series he has constructed does.

 

[morphism]

Try to prove this: Let G be a finite group. If G has a solvable series where the quotients are all abelian, then G has a solvable series where the quotients are all cyclic.