You might be interested in the result that if *n* is odd, |*G*| = *n* for a finite group *G*, and if every subgroup of *G* is normal, then *G* is abelian. (This does not hold if the hypothesis that *n* is odd is ommitted as the quaternion group of order 8 demonstrates.)

A group whose every subgroup is normal is called a *Dedekind group*. A non-abelian Dedekind group is called a *Hamiltonian group*. With this terminology the result simply states that a Dedekind group of odd order is abelian.

The proof is not immediately obvious. It relies on a classification result that states that every Hamiltonian group is a direct product of the quaternion group of order 8, an elemetary abelian 2-group, and a periodic abelian group of odd order. Once this classification result is established, however, the result can be seen easily.