Can you show that all groups of order \(p^n\) for \(p\) prime are solvable? Do you know how to do this for groups of order \(p^r q^s\)?
Solution
1. Every finite \(p\)-group is solvable.
We argue by induction on \(|G|\). The trivial group is solvable. Let \(G\) be a nontrivial finite \(p\)-group. Its center \(Z(G)\) is nontrivial. Hence \[ 1<|Z(G)|<|G| \] unless \(G\) is abelian, in which case \(G\) is already solvable.
The quotient \(G/Z(G)\) is a smaller \(p\)-group, so by induction it is solvable. The center \(Z(G)\) is abelian, hence solvable. An extension of a solvable group by a solvable group is solvable, so \(G\) is solvable.
2. More generally, every finite group of order\[ p^r q^s \]with \(p,q\) prime is solvable.
This is Burnside’s \(p^a q^b\) theorem, whose classical proof uses character theory. Applying Burnside’s theorem with \(a=r\) and \(b=s\) gives the result.