Group Theory/Commutators, solvable and nilpotent groups

commutator Let be a group and let . Then the commutator of and is defined to be the element . {{definition|commutator commutators form a subgroup Let be a group. Then the set forms a subgroup of . By the subgroup criterion, it is sufficient to show that for , the element is of the form for suitable . Indeed, commutator subgroup Let be a group. Then the commutator subgroup of is defined to be . perfect A group is called perfect if and only if . subdirect normal product of perfect groups is direct Let be perfect groups, and let be a subdirect product which is simultaneously a normal subgroup of their outer direct product. Then in fact . It suffices to show that whenever and , then , since is a perfect group. Thus, let be arbitrary, and pick , where for all , such that . Since is a subgroup and normal, the element is in . solvable group is solvable iff maximal normal subgroup is solvable Let be a group, and let be a maximal normal subgroup. Then is solvable if and only if is solvable.