91Ó°ÊÓ

Show that the dihedral groups \(D_{n}, n \geq 3,\) are solvable.

Find a composition series for the indicated group. In each case find the composition factors. $$ S_{4} $$

$$ \text { Show that } \mathbb{Z} \text { does not have a composition series. } $$

Show that if \(G\) is a solvable group, then it has a subnormal series \(G=G_{0} \geq G_{1} \geq G_{2} \geq \ldots \geq G_{n}=\\{e\\}\) in which the factors \(G_{i} / G_{i+1}\) are cyclic groups of prime order for all \(0 \leq i \leq n-1 .\)

Show that if \(G\) is nilpotent, then \(G / Z(G)\) is also nilpotent.