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

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The discussion centers on the correctness of the solution to Problem 2 from the Spring 2007 Algebra Qualifier Exam, specifically regarding the necessary condition for solvable series in finite groups. The author critiques the solution for failing to address that the constructed series must have cyclic quotient groups. The key assertion is that if a finite group G has a solvable series with abelian quotients, then it can be shown that G also has a solvable series with cyclic quotients.

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mrbohn1
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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.
 
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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.
 

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