Exercises

1. 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.)

2. Prove that a finite group is solvable if and only if the factors of a composition series are cyclic groups having prime orders.

3. Prove that if $G$ is a group which has a composition series, then any normal subgroup of $G$ and any factor group of $G$ also have composition series with factors isomorphic to composition factors of $G$.

   (HINT: Mimic the proof of Theorem 11.2.5.)

4. Prove that $N$ is a maximal normal subgroup of $G$ if and only if $G/N$ is simple.

5. Optional Problem Identify the last statement in this section. State where it came from, what it means, and its significance.

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