Editing Commutator subgroup (section)

=== Classes of groups ===
A group <math>G</math> is an '''[[abelian group]]''' if and only if the derived group is trivial: [''G'',''G''] = {''e''}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization.

A group <math>G</math> is a '''[[perfect group]]''' if and only if the derived group equals the group itself: [''G'',''G''] = ''G''. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.

A group with <math>G^{(n)}=\{e\}</math> for some ''n'' in '''N''' is called a '''[[solvable group]]'''; this is weaker than abelian, which is the case ''n'' = 1.

A group with <math>G^{(n)} \neq \{e\}</math> for all ''n'' in '''N''' is called a '''non-solvable group'''.

A group with <math>G^{(\alpha)}=\{e\}</math> for some [[ordinal number]], possibly infinite, is called a '''[[perfect radical|hypoabelian group]]'''; this is weaker than solvable, which is the case ''α'' is finite (a natural number).