- Prove that any infinite abelian group G does not have a composition series.
*(HINT: Suppose it does and come to a contradiction. Also use the result of exercise 4 for Section 6.3.)* - Prove that a finite group is solvable if and only if the factors of a composition series are cyclic groups having prime orders.
- Prove that if is a group which has a composition series, then any normal subgroup of and any factor group of also have composition series with factors isomorphic to composition factors of .
*(HINT: Mimic the proof of Theorem 11.2.5.)* - Prove that is a maximal normal subgroup of if and only if is simple.
- Optional Problem Identify the last statement in this section. State where it came from, what it means, and its significance.

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David Joyner 2007-08-06