Group Theory/Commutators, solvable and nilpotent groups

Definition (commutator):

Let be a group and let . Then the commutator of and is defined to be the element

.

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Proposition (commutators form a subgroup):

Let be a group. Then the set forms a subgroup of .

Proof: By the subgroup criterion, it is sufficient to show that for , the element is of the form for suitable . Indeed,

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Definition (commutator subgroup):

Let be a group. Then the commutator subgroup of is defined to be .

Definition (perfect):

A group is called perfect if and only if .

Proposition (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 .

Proof: 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 .

Definition (solvable):

Proposition (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.