A group is called perfect if . (I.e., is its own commutator subgroup.)
- Prove that every nonabelian simple group is perfect.
- Prove that if are perfect, then is perfect. Extend this to arbitrary joins of perfect groups.
- Prove that any conjugate of a perfect group is perfect.
- Prove that any group has a unique maximal perfect subgroup and that this subgroup is normal.
- Let be a nonabelian simple group. Now is characteristic in , hence normal. Moreover since is nonabelian. Thus , and is perfect.
- Let be perfect subgroups. It follows from the definition of commutator that if , then . Thus . Thus , so that is perfect.
In general, is the -least subgroup containing . Thus if each is perfect, we have for each , so that . Hence is perfect.
- Let be a perfect subgroup. Now . Now . Thus every conjugate of is perfect. (Alternately, conjugate subgroups are isomorphic.)
- Let be a group and let be the set of all perfect subgroups of . By part 2 above, is a perfect subgroup of . Moreover, by definition is maximal, since if is perfect then , so that no subgroup which is strictly -larger than is perfect. Finally, is unique since if is some other -maximal perfect subgroup then , hence .
Now every conjugate of is also a maximal perfect subgroup, as otherwise contradicts the maximalness of . Thus for all . Hence is normal.