Finite $p$-groups are solvable

Induct on \(\abs G\); the trivial group is solvable. A nontrivial finite \(p\)-group \(G\) has nontrivial center \(Z(G)\) by the class equation. The quotient \(G/Z(G)\) is a \(p\)-group of smaller order, hence solvable by induction, and \(Z(G)\) is an abelian normal subgroup. Pulling back a normal series of \(G/Z(G)\) with abelian factors to \(G\) and appending \(Z(G)\trianglerighteq\theset{e}\) gives a normal series of \(G\) with abelian factors.