the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...

commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the...

characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center...

G^{(2)}\triangleright \cdots ,} where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup of G. These two definitions...

{\displaystyle \textstyle \prod _{i=1}^{n}h_{i}} in H/H′, where H′ is the commutator subgroup of H. The order of the factors is irrelevant since H/H′ is abelian...

group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients...

Commutator subgroup – Smallest normal subgroup by which the quotient is commutative Abelianization – Quotienting a group by its commutator subgroup Dihedral...

related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the commutator subgroup of GL, and...

ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [am, bn]...

and the commutator subgroup [ G , G ] {\displaystyle [G,G]} . More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal...