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If H is normal in a group G, where G has a composition series, then G has a composition series one of whose terms is H.

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Hint: Find a composition series for $H$ and one for $G/H$ and then try to glue them (using the fact that you probably know that there is a relationship between the set of normal subgroups of $G/H$ and the set of normal subgroups of $G$ which contain $H$...)