Let be an odd prime and let be a -group. Prove that if every subgroup of is normal then is abelian. (Note that is a nonabelian 2-group with this property, so the result fails for .)
[Disclaimer: I looked at Alfonso Gracia-Saz’s notes when solving this problem.]
We proceed by induction on such that .
For the base case, if , then is cyclic and thus abelian.
For the inductive step, suppose the conclusion holds for all such that , where . Let be a group of order . If is cyclic, then is abelian. If is not cyclic, then by the previous exercise there exists a normal subgroup with . Choose such that . By the hypothesis, are normal. Now and are groups of order , and by the Lattice Isomorphism Theorem and the induction hypothesis, every subgroup of and is normal; thus and are abelian. By this previous exercise, is abelian.